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Self-orthogonal codes are a significant class of linear codes in coding theory and have attracted a lot of attention. In \cite{HLL2023Te,LH2023Se}, $p$-ary self-orthogonal codes were constructed by using $p$-ary weakly regular bent functions, where $p$ is an odd prime. In \cite{WH2023Se}, two classes of non-degenerate quadratic forms were used to construct $q$-ary self-orthogonal codes, where $q$ is a power of a prime. In this paper, we construct new families of $q$-ary self-orthogonal codes using vectorial dual-bent functions. Some classes of at least almost optimal linear codes are obtained from the dual codes of the constructed self-orthogonal codes. In some cases, we completely determine the weight distributions of the constructed self-orthogonal codes. From the view of vectorial dual-bent functions, we illustrate that the works on constructing self-orthogonal codes from $p$-ary weakly regular bent functions \cite{HLL2023Te,LH2023Se} and non-degenerate quadratic forms with $q$ being odd \cite{WH2023Se} can be obtained by our results. We partially answer an open problem on determining the weight distribution of a class of self-orthogonal codes given in \cite{LH2023Se}. As applications, we construct new infinite families of at least almost optimal $q$-ary linear complementary dual codes (for short, LCD codes) and quantum codes.

Association schemes play an important role in algebraic combinatorics and have important applications in coding theory, graph theory and design theory. The methods to construct association schemes by using bent functions have been extensively studied. Recently, in [13], {\"O}zbudak and Pelen constructed infinite families of symmetric association schemes of classes $5$ and $6$ by using ternary non-weakly regular bent functions.They also stated that constructing $2p$-class association schemes from $p$-ary non-weakly regular bent functions is an interesting problem, where $p>3$ is an odd prime. In this paper, using non-weakly regular bent functions, we construct infinite families of symmetric association schemes of classes $2p$, $(2p+1)$ and $\frac{3p+1}{2}$ for any odd prime $p$. Fusing those association schemes, we also obtain $t$-class symmetric association schemes, where $t=4,5,6,7$. In addition, we give the sufficient and necessary conditions for the partitions $P$, $D$, $T$, $U$ and $V$ (defined in this paper) to induce symmetric association schemes.

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code length. Specifically, we first obtain two classes of $q$-ary cyclic Singleton-optimal $(n, k, d=6;r=2)$-LRCs with length $n=3(q+1)$ when $3 \mid (q-1)$ and $q$ is even, and length $n=\frac{3}{2}(q+1)$ when $3 \mid (q-1)$ and $q \equiv 1(\bmod~4)$, respectively. To the best of our knowledge, this is the first construction of $q$-ary cyclic Singleton-optimal LRCs with length $n>q+1$ and minimum distance $d \geq 5$. On the other hand, an LRC acheiving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of $q$-ary perfect LRCs with length $n=\frac{q^m-1}{q-1}$, minimum distance $d=5$ and locality $r=2$.

It is known that partial spreads is a class of bent partitions. In \cite{AM2022Be,MP2021Be}, two classes of bent partitions whose forms are similar to partial spreads were presented. In \cite{AKM2022Ge}, more bent partitions $\Gamma_{1}, \Gamma_{2}, \Gamma_{1}^{\bullet}, \Gamma_{2}^{\bullet}, \Theta_{1}, \Theta_{2}$ were presented from (pre)semifields, including the bent partitions given in \cite{AM2022Be,MP2021Be}. In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions. For any prime $p$, we show that one can generate certain bent partitions (called bent partitions satisfying Condition $\mathcal{C}$) from certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when $p$ is an odd prime, we show that bent partitions satisfying Condition $\mathcal{C}$ one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that $\Gamma_{1}, \Gamma_{2}, \Gamma_{1}^{\bullet}, \Gamma_{2}^{\bullet}, \Theta_{1}, \Theta_{2}$ are bent partitions. We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions. We show that any ternary weakly regular bent function $f: V_{n}^{(3)}\rightarrow \mathbb{F}_{3}$ ($n$ even) of $2$-form can generate a bent partition. When such $f$ is weakly regular but not regular, the generated bent partition by $f$ is not coming from a normal bent partition, which answers an open problem proposed in \cite{AM2022Be}. We give a sufficient condition on constructing partial difference sets from bent partitions, and when $p$ is an odd prime, we provide a characterization of bent partitions satisfying Condition $\mathcal{C}$ in terms of partial difference sets.

The Learning with Errors (LWE) problem has been widely utilized as a foundation for numerous cryptographic tools over the years. In this study, we focus on an algebraic variant of the LWE problem called Group ring LWE (GR-LWE). We select group rings (or their direct summands) that underlie specific families of finite groups constructed by taking the semi-direct product of two cyclic groups. Unlike the Ring-LWE problem described in \cite{lyubashevsky2010ideal}, the multiplication operation in the group rings considered here is non-commutative. As an extension of Ring-LWE, it maintains computational hardness and can be potentially applied in many cryptographic scenarios. In this paper, we present two polynomial-time quantum reductions. Firstly, we provide a quantum reduction from the worst-case shortest independent vectors problem (SIVP) in ideal lattices with polynomial approximate factor to the search version of GR-LWE. This reduction requires that the underlying group ring possesses certain mild properties; Secondly, we present another quantum reduction for two types of group rings, where the worst-case SIVP problem is directly reduced to the (average-case) decision GR-LWE problem. The pseudorandomness of GR-LWE samples guaranteed by this reduction can be consequently leveraged to construct semantically secure public-key cryptosystems., Comment: 45 pages

Linear codes with few weights have significant applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. There are a number of methods to construct linear codes, one of which is based on functions. Furthermore, two generic constructions of linear codes from functions called the first and the second generic constructions, have aroused the research interest of scholars. Recently, in \cite{Nian}, Li and Mesnager proposed two open problems: Based on the first and the second generic constructions, respectively, construct linear codes from non-weakly regular plateaued functions and determine their weight distributions. Motivated by these two open problems, in this paper, firstly, based on the first generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions and determine the weight distributions of the constructed codes. Next, based on the second generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions belonging to $\mathcal{NWRF}$ (defined in this paper) and determine the weight distributions of the constructed codes. We also give the punctured codes of these codes obtained based on the second generic construction and determine their weight distributions. Meanwhile, we obtain some optimal and almost optimal linear codes. Besides, by the Ashikhmin-Barg condition, we have that the constructed codes are minimal for almost all cases and obtain some secret sharing schemes with nice access structures based on their dual codes., Comment: 52 pages, 34 tables

Cyclic codes are an important class of linear codes. Bounding the minimum distance of cyclic codes is a long-standing research topic in coding theory, and several well-known and basic results have been developed on this topic. Recently, locally repairable codes (LRCs) have attracted much attention due to their repair efficiency in large-scale distributed storage systems. In this paper, by employing the singleton procedure technique, we first provide a sufficient condition for bounding the minimum distance of cyclic codes with typical defining sets. Secondly, by considering a specific case, we establish a connection between bounds for the minimum distance of cyclic codes and solutions to a system of inequalities. This connection leads to the derivation of new bounds, including some with general patterns. In particular, we provide three new bounds with general patterns, one of which serves as a generalization of the Betti-Sala bound. Finally, we present a generalized lower bound for a special case and construct several families of $(2, \delta)$-LRCs with unbounded length and minimum distance $2\delta$. It turns out that these LRCs are distance-optimal, and their parameters are new. To the best of our knowledge, this work represents the first construction of distance-optimal $(r, \delta)$-LRCs with unbounded length and minimum distance exceeding $r+\delta-1$., Comment: 35 pages

Vectorial dual-bent functions have recently attracted some researchers' interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions and linear codes. In this paper, we further study vectorial dual-bent functions $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$, where $2\leq m \leq \frac{n}{2}$, $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the prime field $\mathbb{F}_{p}$. We give new characterizations of certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A) in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. When $p=2$, we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Furthermore, we characterize certain bent partitions in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. For general vectorial dual-bent functions $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$ with $F(0)=0, F(x)=F(-x)$ and $2\leq m \leq \frac{n}{2}$, we give a necessary and sufficient condition on constructing association schemes. Based on such a result, more association schemes are constructed from vectorial dual-bent functions.

In distributed storage systems, locally repairable codes (LRCs) are designed to reduce disk I/O and repair costs by enabling recovery of each code symbol from a small number of other symbols. To handle multiple node failures, $(r,\delta)$-LRCs are introduced to enable local recovery in the event of up to $\delta-1$ failed nodes. Constructing optimal $(r,\delta)$-LRCs has been a significant research topic over the past decade. In \cite{Luo2022}, Luo \emph{et al.} proposed a construction of linear codes by using unions of some projective subspaces within a projective space. Several new classes of Griesmer codes and distance-optimal codes were constructed, and some of them were proved to be alphabet-optimal $2$-LRCs. In this paper, we first modify the method of constructing linear codes in \cite{Luo2022} by considering a more general situation of intersecting projective subspaces. This modification enables us to construct good codes with more flexible parameters. Additionally, we present the conditions for the constructed linear codes to qualify as Griesmer codes or achieve distance optimality. Next, we explore the locality of linear codes constructed by eliminating elements from a complete projective space. The novelty of our work lies in establishing the locality as $(2,p-2)$, $(2,p-1)$, or $(2,p)$-locality, in contrast to the previous literature that only considered $2$-locality. Moreover, by combining analysis of code parameters and the C-M like bound for $(r,\delta)$-LRCs, we construct some alphabet-optimal $(2,\delta)$-LRCs which may be either Griesmer codes or not Griesmer codes. Finally, we investigate the availability and alphabet-optimality of $(r,\delta)$-LRCs constructed from our modified framework., Comment: 25 pages

Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. In this paper, we present several constructions of $q$-ary linear codes with two or three weights from vectorial dual-bent functions, where $q$ is a power of an odd prime $p$. The weight distributions of the constructed $q$-ary linear codes are completely determined. We illustrate that some known constructions in the literature can be obtained by our constructions. In some special cases, our constructed linear codes can meet the Griesmer bound. Furthermore, based on the constructed $q$-ary linear codes, we obtain secret sharing schemes with interesting access structures.

Constructions of optimal locally repairable codes (LRCs) achieving Singleton-type bound have been exhaustively investigated in recent years. In this paper, we consider new bounds and constructions of Singleton-optimal LRCs with minmum distance $d=6$, locality $r=3$ and minimum distance $d=7$ and locality $r=2$, respectively. Firstly, we establish equivalent connections between the existence of these two families of LRCs and the existence of some subsets of lines in the projective space with certain properties. Then, we employ the line-point incidence matrix and Johnson bounds for constant weight codes to derive new improved bounds on the code length, which are tighter than known results. Finally, by using some techniques of finite field and finite geometry, we give some new constructions of Singleton-optimal LRCs, which have larger length than previous ones.

Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important role in coding theory. In this paper, we study some results on GQC codes over $\mathbb{Z}_4$ including the normalized generating set, the minimum generating set and the normalized generating set of their dual codes. As an application, new $\mathbb{Z}_4$-linear codes and good nonlinear binary codes are constructed from GQC codes over $\mathbb{Z}_4$.

Secret sharing was firstly proposed in 1979 by Shamir and Blakley respectively. To avoid deficiencies of original schemes, researchers presented improvement schemes, among which the multi-secret sharing scheme (MSS) is significant. There are three categories of MSSs, however, we focus on multi-stage secret sharing scheme (MSSS) recovering secrets with any order in this work. By observing inhomogeneous linear recursions (ILRs) in the literature, we conclude a general formula and divide ILRs into two types according to different variables in them. Utilizing these two kinds of ILRs, we propose four verifiable MSSSs with Ajtai's function, which is a lattice-based function. Our schemes have the following advantages. Firstly, our schemes can detect cheat of the dealer and participants, and are multi-use. Secondly, we have several ways to restore secrets. Thirdly, we can turn our schemes into other types of MSSs due to the universality of our method. Fourthly, since we utilize a lattice-based function to mask shares, our schemes can resist the attack from the quantum computer with computational security. Finally, although our schemes need more memory consumption than some known schemes, we need much less time consumption, which makes our schemes more suitable facing limited computing power., Comment: 36 pages,1 figure, 3 tables

Locally repairable codes (LRCs) with $(r,\delta)$ locality were introduced by Prakash \emph{et al.} into distributed storage systems (DSSs) due to their benefit of locally repairing at least $\delta-1$ erasures via other $r$ survival nodes among the same local group. An LRC achieving the $(r,\delta)$ Singleton-type bound is called an optimal $(r,\delta)$ LRC. Constructions of optimal $(r,\delta)$ LRCs with longer code length and determining the maximal code length have been an important research direction in coding theory in recent years. In this paper, we conduct further research on the improvement of maximum code length of optimal $(r,\delta)$ LRCs. For $2\delta+1\leq d\leq 2\delta+2$, our upper bounds largely improve the ones by Cai \emph{et al.}, which are tight in some special cases. Moreover, we generalize the results of Chen \emph{et al.} and obtain a complete characterization of optimal $(r=2, \delta)$-LRCs in the sense of geometrical existence in the finite projective plane $PG(2,q)$. Within this geometrical characterization, we construct a class of optimal $(r,\delta)$ LRCs based on the sunflower structure. Both the construction and upper bounds are better than previous ones., Comment: Submitted to the 2022 IEEE International Symposium on Information Theory (ISIT)

Bent functions $f: V_{n}\rightarrow \mathbb{F}_{p}$ with certain additional properties play an important role in constructing partial difference sets, where $V_{n}$ denotes an $n$-dimensional vector space over $\mathbb{F}_{p}$, $p$ is an odd prime. In \cite{Cesmelioglu1,Cesmelioglu2}, the so-called vectorial dual-bent functions are considered to construct partial difference sets. In \cite{Cesmelioglu1}, \c{C}e\c{s}melio\v{g}lu \emph{et al.} showed that for vectorial dual-bent functions $F: V_{n}\rightarrow V_{s}$ with certain additional properties, the preimage set of $0$ for $F$ forms a partial difference set. In \cite{Cesmelioglu2}, \c{C}e\c{s}melio\v{g}lu \emph{et al.} showed that for a class of Maiorana-McFarland vectorial dual-bent functions $F: V_{n}\rightarrow \mathbb{F}_{p^s}$, the preimage set of the squares (non-squares) in $\mathbb{F}_{p^s}^{*}$ for $F$ forms a partial difference set. In this paper, we further study vectorial dual-bent functions and partial difference sets. We prove that for vectorial dual-bent functions $F: V_{n}\rightarrow \mathbb{F}_{p^s}$ with certain additional properties, the preimage set of the squares (non-squares) in $\mathbb{F}_{p^s}^{*}$ for $F$ and the preimage set of any coset of some subgroup of $\mathbb{F}_{p^s}^{*}$ for $F$ form partial difference sets. Furthermore, explicit constructions of partial difference sets are yielded from some (non)-quadratic vectorial dual-bent functions. In this paper, we illustrate that almost all the results of using weakly regular $p$-ary bent functions to construct partial difference sets are special cases of our results.

This paper presents a key recovery attack on the cryptosystem proposed by Lau and Tan in a talk at ACISP 2018. The Lau-Tan cryptosystem uses Gabidulin codes as the underlying decodable code. To hide the algebraic structure of Gabidulin codes, the authors chose a matrix of column rank $n$ to mix with a generator matrix of the secret Gabidulin code. The other part of the public key, however, reveals crucial information about the private key. Our analysis shows that the problem of recovering the private key can be reduced to solving a multivariate linear system over the base field, rather than solving a multivariate quadratic system as claimed by the authors. Solving the linear system for any nonzero solution permits us to recover the private key. Apparently, this attack costs polynomial time, and therefore completely breaks the cryptosystem.

Using tools developed in a recent work by Shen and the second author, in this paper we carry out an in-depth study on the average decoding error probability of the random matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random matrix ensemble and the error exponent of the variance, which imply a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity., Comment: 4 figures

Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) from Calderbank-Shor-Steane (CSS) construction. In general, it's difficult to determine the required number of maximally entangled states of an AEAQECC, which is associated with the dimension of the intersection of the two corresponding linear codes. Two linear codes are said to be a linear l-intersection pair if their intersection has dimension l. In this paper, all possible linear l-intersection pairs of MDS codes are given. As an application, we give a complete characterization of pure MDS AEAQECCs for all possible parameters.

This paper presents a new technique for disturbing the algebraic structure of linear codes in code-based cryptography. This is a new attempt to exploit Gabidulin codes in the McEliece setting and almost all the previous cryptosystems of this type have been completely or partially broken. To be specific, we introduce the so-called semilinear transformation in coding theory, which is defined through an $\mathbb{F}_q$-linear automorphism of $\mathbb{F}_{q^m}$, then apply them to construct a public key encryption scheme. Our analysis shows that this scheme can resist all the existing distinguisher attacks, such as Overbeck's attack and Coggia-Couvreur attack. Meanwhile, we endow the underlying Gabidulin code with the so-called partial cyclic structure to reduce the public key size. Compared with some other code-based cryptosystems, our proposal has a much more compact representation of public keys. For instance, 2592 bytes are enough for our proposal to achieve the security of 256 bits, almost 403 times smaller than that of Classic McEliece entering the third round of the NIST PQC project.

This paper presents two public key cryptosystems based on the so-called expanded Gabidulin codes, which are constructed by expanding Gabidulin codes over the base field. Exploiting the fast decoder of Gabidulin codes, we propose an efficient algorithm to decode these new codes when the noise vector satisfies a certain condition. Additionally, these new codes have an excellent error-correcting capability because of the optimality of their parent Gabidulin codes. With different masking techniques, we give two encryption schemes by using expanded Gabidulin codes in the McEliece setting. Being constructed over the base field, these two proposals can prevent the existing structural attacks using the Frobenius map. Based on the distinguisher for Gabidulin codes, we propose a distinguisher for expanded Gabidulin codes by introducing the concept of the so-called twisted Frobenius power. It turns out that the public code in our proposals seems indistinguishable from random codes under this distinguisher. Furthermore, our proposals have an obvious advantage in public key representation without using the cyclic or quasi-cyclic structure compared to some other code-based cryptosystems. To achieve the security of 256 bits, for instance, a public key size of 37583 bytes is enough for our first proposal, while around 1044992 bytes are needed for Classic McEliece selected as a candidate of the third round of the NIST PQC project.

In this paper, we study the dual of generalized bent functions $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The dual of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of \cite{Cesmelioglu5}, we obtain an explicit construction of generalized bent functions for which the dual can be generalized bent or not generalized bent. We show that the generalized indirect sum construction method given in \cite{Wang} can provide a secondary construction of generalized bent functions for which the dual can be generalized bent or not generalized bent. By using the knowledge on ideal decomposition in cyclotomic field, we prove that $f^{**}(x)=f(-x)$ if $f$ is a generalized bent function and its dual $f^{*}$ is also a generalized bent function. For any non-weakly regular generalized bent function $f$ which satisfies that $f(x)=f(-x)$ and its dual $f^{*}$ is generalized bent, we give a property and as a consequence, we prove that there is no self-dual generalized bent function $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ if $p\equiv 3 \ (mod \ 4)$ and $n$ is odd. For $p \equiv 1 \ (mod \ 4)$ or $p\equiv 3 \ (mod \ 4)$ and $n$ is even, we give a secondary construction of self-dual generalized bent functions. In the end, we characterize the relations between the generalized bentness of the dual of generalized bent functions and the bentness of the dual of bent functions, as well as the self-duality relations between generalized bent functions and bent functions by the decomposition of generalized bent functions.

Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in CDMA communication systems. In this paper, we first study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in [34], where the Jacobi sums over Galois rings with characteristics of a square of a prime number were discussed. Then, the deterministic construction of codebooks based on the Jacobi sums over Galois rings of arbitrary characteristics is presented, which produces asymptotically optimal codebooks with respect to the Welch bound. In addition, the parameters of the codebooks provided in this paper are new., Comment: arXiv admin note: substantial text overlap with arXiv:2001.04028

Plateaued functions as an extension of bent functions play a significant role in cryptography, coding theory, sequences and combinatorics. In \cite{Mesnager9}, Mesnager \emph{et al.} introduced generalized plateaued functions in order to study plateaued functions in the general context of generalized $p$-ary functions. In this paper, we focus on the constructions of generalized $p$-ary $s$-plateaued functions from $V_{n}$ to $\mathbb{Z}_{p^k}$, where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$, $p$ is a prime, $k\geq 1$ and $n+s$ is even when $p=2$. In particular, when $k=1$, the constructions in this paper are applicable for plateaued functions. Firstly, inspired by the work of Hod\v{z}i\'{c} \emph{et al}. \cite{Hodzic3} for Boolean plateaued functions, we characterize generalized plateaued functions with affine Walsh supports and provide constructions of generalized plateaued functions with (non)-affine Walsh supports by spectral method. When $p=2, k=1$, our constructions of Boolean plateaued functions with (non)-affine Walsh supports provide an answer to the Open Problem 2 proposed in \cite{Hodzic3}. Secondly, based on what we called generalized indirect sum, we give constructions of generalized plateaued functions, which are also applicable for (non)-weakly regular generalized bent functions. In the end, we discuss the constructions of pairwise disjoint spectra generalized plateaued functions with (non)-affine Walsh supports and we present a construction of generalized bent functions by using pairwise disjoint spectra generalized plateaued functions as building blocks.

Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square q, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of q-ary MDS Euclidean self-dual codes. In particular, for large square q, our constructions take up a proportion of generally more than 34% in all the possible lengths of q-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.

Gametes constitute a critical stage of the plant life cycle during which the genome undergoes reprogramming in preparation for embryogenesis. Here, we examined genome-wide distributions of small RNAs in the sperm and egg cells of rice. We found that 24-nt siRNAs, which are a hallmark of RNA-directed DNA methylation (RdDM) in plants, were depleted from heterochromatin boundaries in both gametes relative to vegetative tissues, reminiscent of siRNA patterns in DDM1-type nucleosome remodeler mutants. In sperm cells, 24-nt siRNAs were spread across heterochromatic regions, while in egg cells, 24-nt siRNAs were concentrated at a smaller number of heterochromatic loci throughout the genome, especially at loci which also produced siRNAs in other tissues. In both gametes, patterns of CHH methylation, typically a strong indicator of RdDM, were similar to vegetative tissues, although lower in magnitude. These findings indicate that the small RNA transcriptome undergoes large-scale redistribution in both male and female gametes, which is not correlated with recruitment of DNA methyltransferases in gametes and suggestive of unexplored regulatory activities of gamete small RNAs.

An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) was introduced by Prakash et al. \cite{Prakash2012} for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$-LRCs produced by Gopalan et al. \cite{Gopalan2012}. An $(r, \delta)$-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. \cite{Chen2018} generalized the construction of cyclic $r$-LRCs proposed by Tamo et al. \cite{Tamo2015,Tamo2016} and constructed several classes of optimal $(r, \delta)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of \cite{Chen2018,Chen2019}, this paper first characterizes $(r, \delta)$-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$-LRCs proposed in \cite{Chen2018,Chen2019}, and our method seems more convenient to obtain optimal cyclic $(r, \delta)$-LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively such that $(r+\delta-1)\nmid n$ can be obtained from our method., Comment: 26 pages

The parameters of a $q$-ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a further study on the construction of MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The main idea of our construction is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we consider the evaluation set consists of two disjoint subsets, one of which is based on the trace function, the other one is a union of a subspace and its cosets. Then four new families of MDS Euclidean self-dual codes are constructed. Secondly, we give a simple but useful lemma to ensure that the symmetric difference of two intersecting subsets of finite fields can be taken as the desired evaluation set. Based on this lemma, we generalize our first construction and provide two new families of MDS Euclidean self-dual codes. Finally, by using two multiplicative subgroups and their cosets which have nonempty intersection, we present three generic constructions of MDS Euclidean self-dual codes with flexible parameters. Several new families of MDS Euclidean self-dual codes are explicitly constructed.

A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions is consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are obtained and they have better parameters than previous results in certain region. Moreover, by the Mobius action over finite fields, we give a systematic way to construct self-dual GRS codes with different evaluation points provided any known self-dual GRS codes. Specially, we prove that all the self-dual extended GRS codes over $\mathbb{F}_{q}$ with length $n< q+1$ can be constructed from GRS codes with the same parameters.

Secret sharing was proposed primarily in 1979 to solve the problem of key distribution. In recent decades, researchers have proposed many improvement schemes. Among all these schemes, the verifiable multi-secret sharing (VMSS) schemes are studied sufficiently, which share multiple secrets simultaneously and perceive malicious dealer as well as participants. By pointing out that the schemes presented by Dehkordi and Mashhadi in 2008 cannot detect some vicious behaviors of the dealer, we propose two new VMSS schemes by adding validity check in the verification phase to overcome this drawback. Our new schemes are based on XTR public key system, and can realize $GF(p^{6})$ security by computations in $GF(p^{2})$ without explicit constructions of $GF(p^{6})$, where $p$ is a prime. Compared with the VMSS schemes using RSA and linear feedback shift register (LFSR) public key cryptosystems, our schemes can achieve the same security level with shorter parameters by using trace function. What's more, our schemes are much simpler to operate than those schemes based on Elliptic Curve Cryptography (ECC). In addition, our schemes are dynamic and threshold changeable, which means that it is efficient to implement our schemes according to the actual situation when participants, secrets or the threshold needs to be changed., Comment: 11 pages, 1 figure, 4 tables

Arylsilanes stand as convenient, inexpensive and versatile reagents for org. synthesis. They enjoy the benefits of low toxicity, low cost and high chem. stability, when compared to other organometallic reagents such as organoboron, organomagnesium or organolithium compds. Furthermore, with the advent of novel, efficient C-H silylation methods, utilization of arylsilanes in synthesis warrants further exploration. Despite the extended research on nitroarenes prepn., ipso-nitration of organosilanes has thus far not been reported. Herein, we disclose a one-pot protocol for the ipsonitration of arylsilanes employing readily available reagents, under mild conditions. The scope and limitations of the present methods will be presented and efforts towards a tandem C-H silylation/nitration protocol will also be discussed.

Featuring guest artist Julia Bullard.; Includes schedule of events.; Includes biographical information on guest and faculty artists.; Includes lists of College of Fine Arts administrators, School of Music administrators, and string music faculty.; Includes a list of upcoming School of Music String events of note (for September 2019-March 2020)., Series LXXIV, Number 012., This archival material has been provided for educational purposes. Ball State University Libraries recognizes that some historic items may include offensive content. Our statement regarding objectionable content is available at: https://dmr.bsu.edu/digital/about

Let $\mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and $R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k$ is an integer satisfying $k\geq 2$. For any odd positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and $2$-quasi-cyclic code of length $4n$ over $\mathbb{F}_{2^m}$ derived by every self-dual cyclic code of length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ and a Gray map from $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ onto $\mathbb{F}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ is determined and all distinct self-orthogonal cyclic codes of length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ are listed.

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring $R$ of length $2^sn$ and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and $2$-quasi-cyclic codes over $\mathbb{F}_{2^m}$ of length $2^{s+1}n$ can be obtained from self-dual cyclic code over $R$ of length $2^sn$ and by a Gray map preserving orthogonality and distances from $R$ onto $\mathbb{F}_{2^m}^2$., Comment: arXiv admin note: text overlap with arXiv:1811.11018, arXiv:1907.07107

A verifiable multi-secret sharing (VMSS) scheme enables the dealer to share multiple secrets, and the deception of both participants and the dealer can be detected. After analyzing the security of VMSS schemes proposed by Mashhadi and Dehkordi in 2015, we illustrate that they cannot detect some deception of the dealer. By using nonhomogeneous linear recursion and LFSR public key cryptosystem, we introduce two new VMSS schemes. Our schemes can not only overcome the drawback mentioned above, but also have shorter private/public key length at the same safety level. Besides, our schemes have dynamism., Comment: 21 pages, 1 figures, 4 tables

Let $m$ be an arbitrary positive integer and $D_{8m}$ be a dihedral group of order $8m$, i.e., $D_{8m}=\langle x,y\mid x^{4m}=1, y^2=1, yxy=x^{-1}\rangle$. Left ideals of the dihedral group algebra $\mathbb{F}_2[D_{8m}]$ are called binary left dihedral codes of length $8m$, and abbreviated as binary left $D_{8m}$-codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$-codes. These codes make up an important class of self-dual binary $[8m,4m]$-codes such that the dihedral group $D_{8m}$ is necessary a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left $D_{8m}$-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary $[48,24,12]$-code and an extremal self-dual binary $[56,28,12]$-code from self-dual binary left $D_{48}$-codes and left $D_{56}$-codes respectively.

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q + 1 are also obtained., Comment: Accepted for publication in IEEE Transactions on Information Theory

In this paper, an explicit representation and enumeration for negacyclic codes of length $2^kn$ over the local non-principal ideal ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$ $(u^2=0)$ is provided, where $k, n$ are any positive integers and $n$ is odd. As a corollary, all distinct negacyclic codes of length $2^k$ over $R$ are listed precisely. Moreover, a mass formula for the number of negacyclic codes of length $2^kn$ over $R$ is given and a mistake in [Cryptogr. Commun. (2017) 9: 241--272] is corrected.

In the two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving (LEP) if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. In this paper (Part II), we first consider the design of a family of LEP SLNCs for a fixed rate and a flexible security level. We present a novel and efficient approach for constructing upon an SLNC that exists an LEP SLNC with the same rate and the security level increased by one. Next, we consider the design of a family of LEP SLNCs for a fixed dimension (equal to the sum of rate and security level) and a flexible pair of rate and security level. We propose another novel approach for designing an SLNC such that the same SLNC can be applied for all the rate and security-level pairs with the fixed dimension. Also, two polynomial-time algorithms are developed for efficient implementations of our two approaches, respectively. Furthermore, we prove that both approaches do not incur any penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. Finally, we consider the ultimate problem of designing a family of LEP SLNCs that can be applied to all possible pairs of rate and security level. By combining the construction of a family of LEP SLNCs for a fixed security level and a flexible rate (obtained in Part I) with the constructions of the two families of LEP SLNCs in the current paper in suitable ways, we can obtain a family of LEP SLNCs that can be applied for all possible pairs of rate and security level. Three possible such constructions are presented., Comment: 38 pages

Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. In the current paper (i.e., Part I of the two-part paper), we focus on the problem for a fixed security level and a flexible rate. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. We present an efficient approach for constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same security level and the rate reduced by one. By applying this approach repeatedly, we can obtain a family of local-encoding-preserving SLNCs with a fixed security level and multiple rates. We also develop a polynomial-time algorithm for efficient implementation of this approach. Furthermore, it is proved that the proposed approach incurs no penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. The result in this paper will be used as a building block for efficiently constructing a family of local-encoding-preserving SLNCs for all possible pairs of rate and security level, which will be discussed in the companion paper (i.e., Part II of the two-part paper)., Comment: 32 pages

It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of q-ary quantum MDS codes by using generalized Reed-Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than q/2+1 Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in [1] and [2], hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well., Comment: Accepted for publication in IEEE Transactions on Information Theory

Locally repairable codes with locality $r$ ($r$-LRCs for short) were introduced by Gopalan et al. \cite{1} to recover a failed node of the code from at most other $r$ available nodes. And then $(r,\delta)$ locally repairable codes ($(r,\delta)$-LRCs for short) were produced by Prakash et al. \cite{2} for tolerating multiple failed nodes. An $r$-LRC can be viewed as an $(r,2)$-LRC. An $(r,\delta)$-LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct $q$-ary optimal $(r,\delta)$-LRCs with length much larger than $q$. Surprisingly, Luo et al. \cite{3} presented a construction of $q$-ary optimal $r$-LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of $q$) via cyclic codes. In this paper, inspired by the work of \cite{3}, we firstly construct two classes of optimal cyclic $(r,\delta)$-LRCs with unbounded lengths and minimum distances $\delta+1$ or $\delta+2$, which generalize the results about the $\delta=2$ case given in \cite{3}. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic $(r,\delta)$-LRCs with unbounded length and larger minimum distance $2\delta$. Furthermore, when $\delta=3$, we give another class of optimal cyclic $(r,3)$-LRCs with unbounded length and minimum distance $6$.

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ where $e\geq 4$ be an even integer. Let $\delta,\alpha\in \mathbb{F}_{p^m}^{\times}$. Then the class of $(\delta+\alpha u^2)$-constacyclic codes over $R$ is a significant subclass of constacyclic codes over $R$ of Type 2. For any integer $k\geq 1$, an explicit representation and a complete description for all distinct $(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $np^k$ and their dual codes are given. Moreover, formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct $(\delta+\alpha u^2)$-contacyclic codes over $\mathbb{F}_{p^m}[u]/\langle u^{e}\rangle$ of length $p^k$ and their dual codes are presented precisely.

Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters \[ [[tq, tq-2d+2, d]]_{q} \] for any $1 \leq t \leq q, 2 \leq d \leq \lfloor \frac{tq+q-1}{q+1}\rfloor+1$, and \[ [[t(q+1)+2, t(q+1)-2d+4, d]]_{q} \] for any $1 \leq t \leq q-1, 2 \leq d \leq t+2$ with $(p,t,d) \neq (2, q-1, q)$. Our quantum codes have flexible parameters, and have minimum distances larger than $\frac{q}{2}+1$ when $t > \frac{q}{2}$. Furthermore, it turns out that our constructions generalize and improve some previous results., Comment: 14 pages. Accepted by Finite Fields and Their Applications

Let $m,e$ be positive integers, $p$ a prime number, $\mathbb{F}_{p^m}$ be a finite field of $p^m$ elements and $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ which is a finite chain ring. For any $\omega\in R^\times$ and positive integers $k, n$ satisfying ${\rm gcd}(p,n)=1$, we prove that any $(1+\omega u)$-constacyclic code of length $p^kn$ over $R$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ cyclic codes with length $n$ over $R$ and a $p^k\times p^k$ matrix $A_{p^k}$ over $\mathbb{F}_p$. Using the matrix-product structures, we give an iterative construction of every $(1+\omega u)$-constacyclic code by $(1+\omega u)$-constacyclic codes of shorter lengths over $R$., Comment: arXiv admin note: text overlap with arXiv:1705.08819

For any prime number $p$, positive integers $m, k, n$ satisfying ${\rm gcd}(p,n)=1$ and $\lambda_0\in \mathbb{F}_{p^m}^\times$, we prove that any $\lambda_0^{p^k}$-constacyclic code of length $p^kn$ over the finite field $\mathbb{F}_{p^m}$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ $\lambda_0$-constacyclic codes with length $n$ over $\mathbb{F}_{p^m}$.

An $(n,k,r)$ \emph{locally repairable code} (LRC) is an $[n,k,d]$ linear code where every code symbol can be repaired from at most $r$ other code symbols. An LRC is said to be optimal if the minimum distance attains the Singleton-like bound $d \le n-k-\lceil k/r \rceil +2$. The \emph{generalized Hamming weights} (GHWs) of linear codes are fundamental parameters which have many useful applications. Generally it is difficult to determine the GHWs of linear codes. In this paper, we study the GHWs of LRCs. Firstly, we obtain a generalized Singleton-like bound on the $i$-th $(1\le i \le k)$ GHWs of general $(n,k,r)$ LRCs. Then, it is shown that for an optimal $(n,k,r)$ LRC with $r \mid k$, its weight hierarchy can be completely determined, and the $i$-th GHW of an optimal $(n,k,r)$ LRC with $r \mid k$ attains the proposed generalized Singleton-like bound for all $1\le i \le k$. For an optimal $(n,k,r)$ LRC with $r \nmid k$, we give lower bounds on the GHWs of the LRC and its dual code. Finally, two general bounds on linear codes in terms of the GHWs are presented. Moreover, it is also shown that some previous results on the bounds of minimum distances of linear codes can also be explained or refined in terms of the GHWs., Comment: 8 pages

We report a novel hollow porous hierarchical-architectured 0.5Li2MnO3·0.5LiMn0.4Co0.3Ni0.3O2 (LLO) for lithium-ion batteries (LIBs). The obtained lithium-rich layered oxides possess a large inner cavity, a permeable porous shell, and excellent structural robustness. In LIBs, such unique features are favorable for fast Li+ transportation and can provide sufficient contact between active materials and electrolytes, accommodate more Li+, and improve the kinetics of the electrochemical reaction. The as-prepared LLO displays an extremely high initial discharge capacity (296.5 mAh g-1 at 0.2 C), high rate capability (162.6 mAh g-1 at 10 C), and excellent cycling stability (237.6 mAh g-1 after 100 cycles at 0.5 C and 153.8 mAh g-1 after 200 cycles at 10 C). These values are superior to most literature data. © 2016 American Chemical Society.

We report a novel hollow porous hierarchical-architectured 0.5Li2MnO3·0.5LiMn0.4Co0.3Ni0.3O2 (LLO) for lithium-ion batteries (LIBs). The obtained lithium-rich layered oxides possess a large inner cavity, a permeable porous shell, and excellent structural robustness. In LIBs, such unique features are favorable for fast Li+ transportation and can provide sufficient contact between active materials and electrolytes, accommodate more Li+, and improve the kinetics of the electrochemical reaction. The as-prepared LLO displays an extremely high initial discharge capacity (296.5 mAh g-1 at 0.2 C), high rate capability (162.6 mAh g-1 at 10 C), and excellent cycling stability (237.6 mAh g-1 after 100 cycles at 0.5 C and 153.8 mAh g-1 after 200 cycles at 10 C). These values are superior to most literature data. © 2016 American Chemical Society.