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Dillon-like Boolean functions are known, in the literature, to be those trace polynomial functions from $\mathbb{F}_{2^{2n}}$ to $\mathbb{F}_{2}$, with all the exponents being multiples of $2^n-1$ often called Dillon-like exponents. This paper is devoted to bent functions in which we study the bentness of some classes of Dillon-like Boolean functions connected with rational trace functions. Specifically, we introduce a special infinite family of trace rational functions. We shall use these functions as building blocks and generalise notably a criterion due to Li et al. published in [IEEE Trans. Inf. Theory 59(3), pp. 1818-1831, 2013] on the bentness of Dillon-like functions in the binary case, we explicitly characterize three classes of bent functions. These characterizations are expressed in terms of the well-known binary Kloosterman sums. Furthermore, analysis and experiments indicate that new functions not EA-equivalent to all known classes of monomial functions are included in our classes., Comment: 28 pages

Let $q$ be an odd prime power with $q\equiv 3\ ({\rm{mod}}\ 4)$. In this paper, we study the differential and boomerang properties of the function $F_{2,u}(x)=x^2\big(1+u\eta(x)\big)$ over $\mathbb{F}_{q}$, where $u\in\mathbb{F}_{q}^*$ and $\eta$ is the quadratic character of $\mathbb{F}_{q}$. We determine the differential uniformity of $F_{2,u}$ for any $u\in\mathbb{F}_{q}^*$ and determine the differential spectra and boomerang uniformity of the locally-APN functions $F_{2,\pm 1}$, thereby disproving a conjecture proposed in \cite{budaghyan2024arithmetization} which states that there exist infinitely many $q$ and $u$ such that $F_{2,u}$ is an APN function.

The Feistel Boomerang Connectivity Table ($\rm{FBCT}$), which is the Feistel version of the Boomerang Connectivity Table ($\rm{BCT}$), plays a vital role in analyzing block ciphers' ability to withstand strong attacks, such as boomerang attacks. However, as of now, only four classes of power functions are known to have explicit values for all entries in their $\rm{FBCT}$. In this paper, we focus on studying the FBCT of the power function $F(x)=x^{2^{n-2}-1}$ over $\mathbb{F}_{2^n}$, where $n$ is a positive integer. Through certain refined manipulations to solve specific equations over $\mathbb{F}_{2^n}$ and employing binary Kloosterman sums, we determine explicit values for all entries in the $\rm{FBCT}$ of $F(x)$ and further analyze its Feistel boomerang spectrum. Finally, we demonstrate that this power function exhibits the lowest Feistel boomerang uniformity.

Let $q = p^m$, where $p$ is an odd prime number and $m$ is a positive integer. In this paper, we examine the finite field $\mathbb{F}_{q^2}$, which consists of $q^2$ elements. We first present an alternative method to determine the differential spectrum of the power function $f(x) = x^{q+2}$ on $\mathbb{F}_{q^2}$, incorporating several key simplifications. This methodology provides a new proof of the results established by Man, Xia, Li, and Helleseth in Finite Fields and Their Applications 84 (2022), 102100, which not only completely determine the differential spectrum of $f$ but also facilitate the analysis of its boomerang uniformity. Specifically, we determine the boomerang uniformity of $f$ for the cases where $q \equiv 1$ or $3$ (mod $6$), with the exception of the scenario where $p = 5$ and $m$ is even. Furthermore, for $p = 3$, we investigate the value distribution of the Walsh spectrum of $f$, demonstrating that it takes on only four distinct values. Using this result, we derive the weight distribution of a ternary cyclic code with four Hamming weights. The article integrates refined mathematical techniques from algebraic number theory and the theory of finite fields, employing several ingredients, such as exponential sums, to explore the cryptographic analysis of functions over finite fields. They can be used to explore the differential/boomerang uniformity across a wider range of functions.

In this paper, we investigate permutation polynomials over a finite field $\mathbb F_{q^n}$ with $q=2^m$, focusing on those in the form $\mathrm{Tr}(Ax^{q+1})+L(x)$, where $A\in\mathbb F_{q^n}^*$ and $L$ is a $2$-linear polynomial over $\mathbb F_{q^n}$. By calculating certain character sums, we characterize these permutation polynomials and provide additional constructions.

Audit logs are one of the most important tools for transparently tracking system events and maintaining continuous oversight in corporate organizations and enterprise business systems. There are many cases where the audit logs contain sensitive data, or the audit logs are enormous. In these situations, dealing with a subset of the data is more practical than the entire data set. To provide a secure solution to handle these issues, a sanitizable signature scheme (SSS) is a viable cryptographic primitive. Herein, we first present the first post-quantum secure multivariate-based SSS, namely Mul-SAN. Our proposed design provides unforgeability, privacy, immutability, signer accountability, and sanitizer accountability under the assumption that the MQ problem is NP-hard. Mul-SAN is very efficient and only requires computing field multiplications and additions over a finite field for its implementation. Mul-SAN presents itself as a practical method to partially delegate control of the authenticated data in avenues like the healthcare industry and government organizations. We also explore using Blockchain to provide a tamper-proof and robust audit log mechanism.

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results have been obtained intensively for LCD codes, but only partial results have been derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterisation and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes $(\mathcal{C}, \mathcal{D})$ (notably for cyclic codes), which are given by the minimum distances $d(\mathcal{C})$ and $d(\mathcal{D}^\perp)$. Further, we show that for LCP algebraic geometry codes $(\mathcal{C},\mathcal{D})$, the dual code $\mathcal{C}^\perp$ is equivalent to $\mathcal{D}$ under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in [``Complementary dual algebraic geometry codes", IEEE Trans. Inform Theory, vol. 64(4), 2390--3297, 2018]. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied.

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special $q$-quadratic polynomial in $\mathbb F_{q^2}$, by calculating character sums related to quadratic forms of $\mathbb F_{q^2}/\mathbb F_q$. Then given some rational function, we can demonstrate whether it induces a permutation of $\mathbb F_{q^2}$.

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates.

Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Difference Distribution Table (DDT), the Feistel Boomerang Connectivity Table (FBCT), the Feistel Boomerang Difference Table (FBDT) and the Feistel Boomerang Extended Table (FBET) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, the results on them are rare. In this paper, we investigate the properties of the power function $F(x):=x^{2^{m+1}-1}$ over the finite field $\gf_{2^n}$ of order $2^n$ where $n=2m$ or $n=2m+1$ ($m$ stands for a positive integer). As a consequence, by carrying out certain finer manipulations of solving specific equations over $\gf_{2^n}$, we give explicit values of all entries of the DDT, the FBCT, the FBDT and the FBET of the investigated power functions. From the theoretical point of view, our study pushes further former investigations on differential and Feistel boomerang differential uniformities for a novel power function $F$. From a cryptographic point of view, when considering Feistel block cipher involving $F$, our in-depth analysis helps select $F$ resistant to differential attacks, Feistel differential attacks and Feistel boomerang attacks, respectively.

Subspace codes have important applications in random network coding. It is interesting to construct subspace codes with both sizes, and the minimum distances are as large as possible. In particular, cyclic constant dimension subspaces codes have additional properties which can be used to make encoding and decoding more efficient. In this paper, we construct large cyclic constant dimension subspace codes with minimum distances $2k-2$ and $2k$. These codes are contained in $\mathcal{G}_q(n, k)$, where $\mathcal{G}_q(n, k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{F}_{q^n}$. Consequently, some results in \cite{FW}, \cite{NXG}, and \cite{ZT} are extended., Comment: This article was submitted to Designs, Codes and Cryptography on November 22nd, 2022

A substitution box (S-box) in a symmetric primitive is a mapping $F$ that takes $k$ binary inputs and whose image is a binary $m$-tuple for some positive integers $k$ and $m$, which is usually the only nonlinear element of the most modern block ciphers. Therefore, employing S-boxes with good cryptographic properties to resist various attacks is significant. For power permutation $F$ over finite field $\GF{2^k}$, the multiset of values $\beta_F(1,b)=\#\{x\in \GF{2^k}\mid F^{-1}(F(x)+b)+F^{-1}(F(x+1)+b)=1\}$ for $b\in \GF{2^k}$ is called the boomerang spectrum of $F$. The maximum value in the boomerang spectrum is called boomerang uniformity. This paper determines the boomerang spectrum of the power permutation $X^{2^{3n}+2^{2n}+2^{n}-1}$ over $\GF{2^{4n}}$. The boomerang uniformity of that power permutation is $3(2^{2n}-2^n)$. However, on a large subset $\{b\in \GF{2^{4n}}\mid \mathbf{Tr}_n^{4n}(b)\neq 0\}$ of $\GF{2^{4n}}$ of cardinality $2^{4n}-2^{3n}$ (where $ \mathbf{Tr}_n^{4n}$ is the (relative) trace function from $\GF{2^{4n}}$ to $\GF{2^{n}}$), we prove that the studied function $F$ achieves the optimal boomerang uniformity $2$. It is known that obtaining such functions is a challenging problem. More importantly, the set of $b$'s giving this value is explicitly determined for any value in the boomerang spectrum.

Let $F(X)=X^{2^{2k}+2^k+1}$ be the power function over the finite field $\GF{2^{4k}}$ which is known as the Bracken-Leander function. In \cite{BCC10,BL10,CV20,Fu22,XY17}, it was proved that the number of solutions in $\GF{q^4}$ to the equation $F(X)+F(X+1)=b$ is in $\{0,2,4\}$ for any $b\in \GF{q^4}$ and the number of the $b$ giving $i$ solutions have been determined for every $i$. However, no paper provided a direct and complete method to solve such an equation, and this problem remained open. This article presents a direct technique to derive an explicit solution to that equation. The main result in \cite{BCC10,BL10,Fu22,XY17}, determining differential spectrum of $F(X)=X^{2^{2k}+2^k+1}$ over $\GF{2^{4k}}$, is re-derived simply from our results.

This paper deals with Niho functions which are one of the most important classes of functions thanks to their close connections with a wide variety of objects from mathematics, such as spreads and oval polynomials or from applied areas, such as symmetric cryptography, coding theory and sequences. In this paper, we investigate specifically the $c$-differential uniformity of the power function $F(x)=x^{s(2^m-1)+1}$ over the finite field $\mathbb{F}_{2^n}$, where $n=2m$, $m$ is odd and $s=(2^k+1)^{-1}$ is the multiplicative inverse of $2^k+1$ modulo $2^m+1$, and show that the $c$-differential uniformity of $F(x)$ is $2^{\gcd(k,m)}+1$ by carrying out some subtle manipulation of certain equations over $\mathbb{F}_{2^n}$. Notably, $F(x)$ has a very low $c$-differential uniformity equals $3$ when $k$ and $m$ are coprime.

We introduce a class of permutation polynomial over $\mathbb F_{q^n}$ that can be written in the form $\frac{L(x)}{x^{q+1}}$ or $\frac{L(x^{q+1})}x$ for some $q$-linear polynomial $L$ over $\mathbb F_{q^n}$. Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form.

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ on an extension of the finite field $\mathbb F_{q^n}/\mathbb F_q$. Specifically, we investigate those functions on $\mathbb F_{q^2}/\mathbb F_q$ and construct several typical kinds of planar functions. We also completely characterize them on $\mathbb F_{q^3}/\mathbb F_q$. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.

Planar functions, introduced by Dembowski and Ostrom, are functions from a finite field to itself that give rise to finite projective planes. They exist, however, only for finite fields of odd characteristics. They have attracted much attention in the last decade thanks to their interest in theory and those deep and various applications in many fields. This paper focuses on planar trinomials over cubic and quartic extensions of finite fields. Our achievements are obtained using connections with quadratic forms and classical algebraic tools over finite fields. Furthermore, given the generality of our approach, the methodology presented could be employed to drive more planar functions on some finite extension fields.

S-boxes are an important primitive that help cryptographic algorithms to be resilient against various attacks. The resilience against specific attacks can be connected with a certain property of an S-box, and the better the property value, the more secure the algorithm. One example of such a property is called boomerang uniformity, which helps to be resilient against boomerang attacks. How to construct S-boxes with good boomerang uniformity is not always clear. There are algebraic techniques that can result in good boomerang uniformity, but the results are still rare. In this work, we explore the evolution of S-boxes with good values of boomerang uniformity. We consider three different encodings and five S-box sizes. For sizes $4\times 4$ and $5\times 5$, we manage to obtain optimal solutions. For $6\times 6$, we obtain optimal boomerang uniformity for the non-APN function. For larger sizes, the results indicate the problem to be very difficult (even more difficult than evolving differential uniformity, which can be considered a well-researched problem)., Comment: 15 pages, 3 figures, 4 tables

Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also analyzed. Some of the resultant $q$-ary codes $C_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes $C_{f,g}$ are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternary Hermitian self-dual code are obtained with $m \geq 2$. As an application, we show that three families of the derived linear codes give rise to several infinite families of $t$-designs ($t \in \{2, 3\}$)., Comment: arXiv admin note: text overlap with arXiv:1804.06003, arXiv:2207.07262 by other authors

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles [Discrete Math. 321, 2014] and [SIAM J. Discrete Math. 27(4), 2013], Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall provide answers to some open problems presented in the literature. The first one concerns Open Problem 1, proposed by Ding and Zhou in Discrete Math. 321, 2014. The two others are Open Problems 5.16 and 5.25, raised by Ding in [SIAM J. Discrete Math. 27(4), 2013].

Only three classes of Almost Perfect Nonlinear (for short, APN) power functions over odd characteristic finite fields have been investigated in the literature, and their differential spectra were determined. The differential uniformity of the power function $F(x)=x^{\frac{p^{n}-3}{2}}$ over the finite field $F_{p^n}$ of order $p^n$ (where $p$ is an odd prime), was studied by Helleseth and Sandberg in 1997, where $p^n\equiv3\pmod{4}$ is an odd prime power with $p^n>7$. It was shown that $F$ is PN when $p^n=27$, APN when $5$ is a nonsquare in $F_{p^n}$, and differentially $3$-uniform when $5$ is a square in $F_{p^n}$. In this paper, by investigating some equation systems and certain character sums over $F_{p^n}$, the differential spectrum of $F$ is completely determined. We focusing on the power functions $x^d$ with even $d$ over $F_{p^n}$ ($p$ odd), the power functions $F$ we consider are APN which are of the lowest differential uniformity and the nontrivial differential spectrum. Moreover, we examine the extension of the so-called $c$-differential uniformity by investigating the $c$-differential properties of $F$. Specifically, an upper bound of the $c$-differential uniformity of $F$ is given, and its $c$-differential spectrum is considered in the case where $c=-1$. Finally, we emphasize that, throughout our study of the differential spectrum of the considered power functions, we provide methods for evaluating sums of specific characters with connections to elliptic curves and for determining the number of solutions of specific systems of equations over finite fields., Comment: arXiv admin note: text overlap with arXiv:2210.09822

Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permutations over finite fields of odd characteristic. Specifically, its objective is twofold concerning the power functions $F(x)=x^{\frac{p^n+3}{2}}$ defined over the finite field $F_{p^n}$ of order $p^n$, where $p$ is an odd prime, and $n$ is a positive integer. The first is to complement some former results initiated by Helleseth and Sandberg in \cite{HS} by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of $F$ when $p^n\equiv3\pmod 4$ and $p\neq 3$. The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some exponential sums over $F_{p^n}$ (which amounts to evaluating the number of $F_{p^n}$-rational points on some related curves and secondly by computing the number of solutions in $(F_{p^n})^4$ of a system of equations presented by Helleseth, Rong, and Sandberg in ["New families of almost perfect nonlinear power mappings," IEEE Trans. Inform. Theory, vol. 45. no. 2, 1999], naturally appears while determining the differential spectrum of $F$. We show that in the considered case ($p^n\equiv3\pmod 4$ and $p\neq 3$), $F$ is an APN power permutation when $p^n=11$, and a differentially $4$-uniform power permutation otherwise.

In this article, we focus on the concept of locally-APN-ness (``APN" is the abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow larger regarding their differential uniformity and, therefore, possibly, more suitable candidates against the differential attack (or their variants). Specifically, given two coprime positive integers $m$ and $k$ such that $\gcd(2^m+1,2^k+1)=1$, we investigate the locally-APN-ness property of an infinite family of Niho type power functions in the form $F(x)=x^{s(2^m-1)+1}$ over the finite field ${\mathbb F}_{2^{2m}}$ for $s=(2^k+1)^{-1}$, where $(2^k+1)^{-1}$ denotes the multiplicative inverse modulo $2^m+1$. By employing finer studies of the number of solutions of certain equations over finite fields (with even characteristic) as well as some subtle manipulations of solving some equations, we prove that $F(x)$ is locally APN and determine its differential spectrum. It is worth noting that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for $2\leq m\leq10$. In addition, we also determine the boomerang spectrum of $F(x)$ by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang, and Li.

This article determines all the solutions in the finite field $GF{2^{4n}}$ of the equation $x^{2^{3n}+2^{2n}+2^{n}-1}+(x+1)^{2^{3n}+2^{2n}+2^{n}-1}=b$. Specifically, we explicitly determine the set of $b$'s for which the equation has $i$ solutions for any positive integer $i$. Such sets, which depend on the number of solutions $i$, are given explicitly and expressed nicely, employing the absolute trace function over $GF{2^{n}}$, the norm function over $GF{2^{4n}}$ relatively to $GF{2^{n}}$ and the set of $2^n+1$st roots of unity in $GF{2^{4n}}$. The equation considered in this paper comes from an article by Budaghyan et al. \cite{BCCDK20}. As an immediate consequence of our results, we prove that the above equation has $2^{2n}$ solutions for one value of $b$, $2^{2n}-2^n$ solutions for $2^n$ values of $b$ in $GF{2^{4n}}$ and has at most two solutions for all remaining points $b$, leading to complete proof of the conjecture raised by Budaghyan et al. We highlight that the recent work of Li et al., in \cite{Li-et-al-2020} gives the complete differential spectrum of $F$ and also gives an affirmative answer to the conjecture of Budaghyan et al. However, we emphasize that our approach is interesting and promising by being different from Li et al. Indeed, on the opposite to their article, our technique allows determine ultimately the set of $b$'s for which the considered equation has solutions as well as the solutions of the equation for any $b$ in $GF{2^{4n}}$.

Aiming to recover the data from several concurrent node failures, linear $r$-LRC codes with locality $r$ were extended into $(r, \delta)$-LRC codes with locality $(r, \delta)$ which can enable the local recovery of a failed node in case of more than one node failure. Optimal LRC codes are those whose parameters achieve the generalized Singleton bound with equality. In the present paper, we are interested in studying optimal LRC codes over small fields and, more precisely, over $\mathbb{F}_4$. We shall adopt an approach by investigating optimal quaternary $(r,\delta)$-LRC codes through their parity-check matrices. Our study includes determining the structural properties of optimal $(r,\delta)$-LRC codes, their constructions, and their complete classification over $\F_4$ by browsing all possible parameters. We emphasize that the precise structure of optimal quaternary $(r,\delta)$-LRC codes and their classification are obtained via the parity-check matrix approach use proofs-techniques different from those used recently for optimal binary and ternary $(r,\delta)$-LRC codes obtained by Hao et al. in [IEEE Trans. Inf. Theory, 2020, 66(12): 7465-7474].

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms., Comment: arXiv admin note: text overlap with arXiv:2010.09448

Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase position and end-shift position, also having zero cross-correlation sums around the end-shift position. It can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio $(\mathrm{CZC}_{\mathrm{ratio}})$ by exploring two promising approaches. The first one of CZCPs via properly cascading sequences from a Golay complementary pair (GCP). The proposed construction leads to $(28L,13L)-\mathrm{CZCPs}$, $(28L,13L+\frac{L}{2})-\mathrm{CZCPs}$ and $(30L,13L-1)-\mathrm{CZCPs}$, where $L$ is the length of a binary GCP. Besides, we emphasize that, our proposed CZCPs have the largest $\mathrm{CZC}_{\mathrm{ratio}}=\frac{27}{28}$, compared with known CZCPs but no-perfect CZCPs in the literature. Specially, we proposed optimal binary CZCPs with $(28,13)-\mathrm{CZCP}$ and $(56,27)-\mathrm{CZCP}$. The second one of CZCPs based on Boolean functions (BFs), and the construction of CZCPs have the largest $\mathrm{CZC}_{\mathrm{ratio}}=\frac{13}{14}$, compared with known CZCPs but no-perfect CZCPs in the literature.

This paper is devoted to sequences and focuses on designing new two-dimensional (2-D) Z-complementary array pairs (ZCAPs) by exploring two promising approaches. A ZCAP is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all time shifts in a zone around the $(0,0)$ time shift, except the $(0,0)$ time shift. The first approach investigated in this paper uses a one-dimensional (1-D) Z-complementary pair (ZCP), which is an extension of the 1-D Golay complementary pair (GCP) where the autocorrelations of constituent sequences are complementary within a zero correlation zone (ZCZ). The second approach involves directly generalized Boolean functions (which are important components with many applications, particularly in (symmetric) cryptography). Along with this paper, new construction of 2-D ZCAPs is proposed based on 1-D ZCP, and direct construction of 2-D ZCAPs is also offered directly by 2-D generalized Boolean functions. Compared to existing constructions based on generalized Boolean functions, our proposed construction covers all of them. ZCZ sequences are a class of spreading sequences having ideal auto-correlation and cross-correlation in a zone around the origin. In recent years, they have been extensively studied due to their crucial applications, particularly in quasi-synchronous code division multiple access systems. Our proposed 2-D ZCAPs based on 2-D generalized Boolean functions have larger 2-D $\mathrm{ZCZ}_{\mathrm{ratio}}=\frac{6}{7}$. Compared to the construction based on ZCPs, our proposed 2-D ZCAPs also have the largest 2-D $\mathrm{ZCZ}_{\mathrm{ratio}}$.

Because of the recent applications to distributed storage systems, researchers have introduced a new class of block codes, i.e., locally recoverable (LRC) codes. LRC codes can recover information from erasure(s) by accessing a small number of erasure-free code symbols and increasing the efficiency of repair processes in large-scale distributed storage systems. In this context, Tamo and Barg first gave a breakthrough by cleverly introducing a good polynomial notion. Constructing good polynomials for locally recoverable codes achieving Singleton-type bound (called optimal codes) is challenging and has attracted significant attention in recent years. This article aims to increase our knowledge of good polynomials for optimal LRC codes. Using tools from algebraic function fields and Galois theory, we continue investigating those polynomials and studying them by developing the Galois theoretical approach initiated by Micheli in 2019. Specifically, we push further the study of a crucial parameter $\mathcal G(f)$ (of a given polynomial $f$), which measures how much a polynomial is "good" in the sense of LRC codes. We provide some characterizations of polynomials with minimal Galois groups and prove some properties of finite fields where polynomials exist with a specific size of Galois groups. We also present some explicit shapes of polynomials with small Galois groups. For some particular polynomials $f$, we give the exact formula of $\mathcal G(f)$.

The problem of solving explicitly the equation $P_a(X):=X^{q+1}+X+a=0$ over the finite field $\GF{Q}$, where $Q=p^n$, $q=p^k$ and $p$ is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \cite{ACZ2000}, the construction of difference sets with Singer parameters \cite{DD2004}, determining cross-correlation between $m$-sequences \cite{DOBBERTIN2006} and to construct error correcting codes \cite{Bracken2009}, cryptographic APN functions \cite{BTT2014,Budaghyan-Carlet_2006}, designs \cite{Tang_2019}, as well as to speed up the index calculus method for computing discrete logarithms on finite fields \cite{GGGZ2013,GGGZ2013+} and on algebraic curves \cite{M2014}. Subsequently, in \cite{Bluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019,KCM19}, the $\GF{Q}$-zeros of $P_a(X)$ have been studied. In \cite{Bluher2004}, it was shown that the possible values of the number of the zeros that $P_a(X)$ has in $\GF{Q}$ is $0$, $1$, $2$ or $p^{\gcd(n, k)}+1$. Some criteria for the number of the $\GF{Q}$-zeros of $P_a(x)$ were found in \cite{HK2008,HK2010,BTT2014,KM2019,MS2019}. However, while the ultimate goal is to explicit all the $\GF{Q}$-zeros, even in the case $p=2$, it was solved only under the condition $\gcd(n, k)=1$ \cite{KM2019}. In this article, we discuss this equation without any restriction on $p$ and $\gcd(n,k)$. In \cite{KCM19}, for the cases of one or two $\GF{Q}$-zeros, explicit expressions for these rational zeros in terms of $a$ were provided, but for the case of $p^{\gcd(n, k)}+1$ $\GF{Q}-$ zeros it was remained open to explicitly compute the zeros. This paper solves the remained problem, thus now the equation $X^{p^k+1}+X+a=0$ over $\GF{p^n}$ is completely solved for any prime $p$, any integers $n$ and $k$., Comment: arXiv admin note: text overlap with arXiv:1912.12648