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Let n be a positive integer. An NFSR of n stages is called irreducible if the family of output sequences of any NFSR of stages less than n is not included in that of the NFSR. In this paper, we prove that the density of the irreducible NFSRs of n stages is larger than 0.39. This implies that it is expected to find an irreducible NFSR of n stages among three randomly chosen NFSRs of n stages. [ABSTRACT FROM PUBLISHER]

In this paper, the sequence families of which maximum correlation is determined by the Weil bound of exponential sums are revisited. Using the same approach, two new constructions with large family sizes and low maximum correlation are given. The first construction is an analog of one recent result derived from the interleaved structure of Sidel'nikov sequences. For a prime p and an integer M\vert (p-1), the new M-ary sequence families of period p are obtained from irreducible quadratic polynomials and known power residue-based sequence families. The new sequence families increase family sizes of the known power residue-based sequence families, but keep the maximum correlation unchanged. In the second construction, the sequences derived from the Weil representation are generalized, where each new sequence is the elementwise product of a modulated Sidel'nikov sequence and a modulated trace sequence. For positive integers d

This paper studies the distinctness of primitive sequences over \bf Z/(M) modulo 2, where M is an odd integer that is composite and square-free, and \bf Z/(M) is the integer residue ring modulo M. A new sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial f\left(x\right) over \bf Z/(M) are pairwise distinct modulo 2. Such result improves a recent result obtained in our previous paper, and consequently, the set of primitive sequences over \bf Z/(M) that can be proven to be distinct modulo 2 is greatly enlarged. [ABSTRACT FROM PUBLISHER]

This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is \ssr NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided \ssr P\ne\ssr NP. [ABSTRACT FROM PUBLISHER]

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with the well-known binary Varshamov–Tenengolts codes, which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions. [ABSTRACT FROM AUTHOR]

Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let ${\mathbb {F}}_{q}$ be a finite field with order $q$ , where $q$ is a positive power of a prime $p$. Suppose that $n$ is a positive integer and the product of distinct prime factors of $n$ divides $q-1$ , i.e., $rad(n)\mid (q-1)$. In this paper, we explicitly factorize the polynomial $x^{n}-\lambda $ for each $\lambda \in {\mathbb {F}}_{q}^{*}$. As applications, first, we obtain all repeated-root $\lambda $ -constacyclic codes and their dual codes of length $np^{s}$ over ${\mathbb {F}}_{q}$ ; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length $n$ over ${\mathbb {F}}_{q}$ ; third, we list all self-dual repeated-root negacyclic codes of length $np^{s}$ over ${\mathbb {F}}_{q}$. In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters. [ABSTRACT FROM AUTHOR]

For a function $f$ from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n}$ , the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an “intermediate” concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it conveys a detailed structural information about $f$. We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges. [ABSTRACT FROM AUTHOR]

Partial MDS (PMDS) codes are a class of erasure-correcting array codes that combine local correction of the rows with global correction of the array. An $\boldsymbol {m}\times \boldsymbol {n}$ array code is called an $(\boldsymbol {r};\boldsymbol {s})$ PMDS code if each row belongs to an ${[}\boldsymbol {n},\boldsymbol {n}-\boldsymbol {r}, \boldsymbol {r}+\textbf {1}{]}$ MDS code and the code can correct erasure patterns consisting of $\boldsymbol {r}$ erasures in each row together with $\boldsymbol {s}$ more erasures anywhere in the array. While a recent construction by Calis and Koyluoglu generates $(\boldsymbol {r};\boldsymbol {s})$ PMDS codes for all $\boldsymbol {r}$ and $\boldsymbol {s}$ , its field size is exponentially large. In this paper, a family of PMDS codes with field size ${\mathcal{ O}}\left ({\max \{\boldsymbol {m},\boldsymbol {n}^{\boldsymbol {r}+\boldsymbol {s}}\}^{\boldsymbol {s}} }\right)$ is presented for the case where $\boldsymbol {r}= {\mathcal{ O}}(1), \boldsymbol {s}= {\mathcal{ O}}(1)$. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained. [ABSTRACT FROM AUTHOR]

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials. [ABSTRACT FROM AUTHOR]

The linear model, in which a set of observations is assumed to be given by a linear combination of columns of a matrix (often termed a dictionary), has long been the mainstay of the statistics and signal processing literature. One particular challenge for inference under linear models is understanding the conditions on the dictionary under which reliable inference is possible. This challenge has attracted renewed attention in recent years, since many modern inference problems (e.g., high-dimensional statistics and compressed sensing) deal with the underdetermined setting, in which the number of observations is much smaller than the number of columns in the dictionary. This paper makes several contributions for this setting when the set of observations is given by a linear combination of a small number of groups of columns of the dictionary, termed the block-sparse case. First, it specifies conditions on the dictionary under which most block submatrices of the dictionary (often termed block subdictionaries) are well conditioned. This result is fundamentally different from prior work on block-sparse inference because: 1) it provides conditions that can be explicitly computed in polynomial time; 2) the given conditions translate into near-optimal scaling of the number of columns of the block subdictionaries as a function of the number of observations for a large class of dictionaries; and 3) it suggests that the spectral norm, rather than the column/block coherences of the dictionary, fundamentally limits the scaling of dimensions of the well-conditioned block subdictionaries. Second, in order to help understand the significance of this result in the context of block-sparse inference, this paper investigates the problems of block-sparse recovery and block-sparse regression in underdetermined settings. In both of these problems, this paper utilizes its result concerning conditioning of block subdictionaries and establishes that near-optimal block-sparse recovery and block-sparse regression is possible for a large class of dictionaries as long as the dictionary satisfies easily computable conditions and the coefficients describing the linear combination of groups of columns can be modeled through a mild statistical prior. Third, the paper reports extensive numerical experiments that highlight the effects of different measures of the dictionary in block-sparse inference problems. [ABSTRACT FROM PUBLISHER]

In this paper, we study the number of different interference alignment (IA) solutions in a $K$ -user multiple-input multiple-output (MIMO) interference channel, when the alignment is performed via beamforming and no symbol extensions are allowed. We focus on the case where the number of IA equations matches the number of variables. In this situation, the number of IA solutions is finite and constant for any channel realization out of a zero-measure set and, as we prove in this paper, it is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of alignment solutions is the scaled average of the determinant of a certain Hermitian matrix related to the geometry of the problem. Interestingly, while the value of this determinant at an arbitrary point can be used to check the feasibility of the IA problem, its average (properly scaled) gives the number of solutions. For single-beam systems, the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented. Nonetheless, our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas, and streams per user. [ABSTRACT FROM PUBLISHER]

Because of their efficient encoding and decoding algorithms, cyclic codes—an interesting class of linear codes—are widely used in communication systems, storage devices, and consumer electronics. BCH codes form a special class of cyclic codes, and are usually among the best cyclic codes. A subclass of good BCH codes is the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The main objective of this paper is to study the dimension and minimum distances of a subclass of the narrow-sense primitive BCH codes with design distance \delta =(q-\ell 0)q^{m-\ell 1-1}-1 for certain pairs (\ell 0, \ell 1) , where 0 \leq \ell 0 \leq q-2 and 0 \leq \ell 1 \leq m-1 . The parameters of other related classes of BCH codes are also investigated, and some open problems are proposed in this paper. [ABSTRACT FROM PUBLISHER]

In this paper, we show that the maximum number of bent component functions of a vectorial function F:GF(2)^n\to GF(2)^n is 2^n-2^n/2 . We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F\in GF(2^n)[x] , where F has only a few terms. The only known power functions having such a large number of bent components are x^{d} , where d=2^{n/2}+1 . In this paper, we show that the binomials F^i(x)=x^2^i(x+x^2^n/2) also have such a large number of bent components, and these binomials are inequivalent to the monomials x^2^n/2+1 if 0

In this paper, the basic function field of an asymptotically optimal tower of function fields by Garcia and Stichtenoth is studied, where bases of one-point linear systems can be explicitly constructed. A representation of finite GF(q^2) -rational points on the associated plane curve is derived in this paper. This representation is exploited to extend the use of Horner’s loops and the mechanism of Chien search in the decoding of Reed–Solomon codes for syndrome generation and error-location search, respectively, to algebraic-geometry codes built on the plane Garcia–Stichtenoth curve. [ABSTRACT FROM AUTHOR]

The most computationally intensive step of the Guruswami–Sudan list decoder for generalized Reed–Solomon codes is the formation of a bivariate interpolation polynomial. Complexity can be reduced if this polynomial has prefactors, i.e., factors of its univariate constituent polynomials that are independent of the received vector, and hence known a priori. For example, the well-known re-encoding projection due to Koetter et al. leads to one class of prefactors. This paper introduces so-called Sierpínski prefactors that result from the property that many binomial coefficients, which arise in the multiplicity constraints defined in terms of the Hasse derivative, are zero modulo the underlying field characteristic. It is shown that re-encoding prefactors and Sierpínski prefactors can be combined to achieve a significantly reduced Guruswami–Sudan interpolation step. In certain practically relevant cases, the introduction of Sierpínski prefactors reduces the number of unknown polynomial coefficients by an additional 10% or more (beyond the reduction due to re-encoding prefactors alone), without incurring additional computational effort at the decoder. [ABSTRACT FROM AUTHOR]

Consider a binary Reed–Muller code RM (r,m) defined on the m -dimensional hypercube \mathbb F2^{m} . In this paper, we study punctured Reed–Muller codes Pr(m,b) , whose positions are restricted to the m -tuples of a given Hamming weight b . In combinatorial terms, this paper concerns m -variate Boolean polynomials of any degree r , which are evaluated on a Hamming sphere of some radius b . Codes Pr(m,b) inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical b -layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes Pr(m,b) and find their distances for the arbitrary values of parameters $r,m$ , and $b$ . Finally, we describe recursive (successive cancellation) decoding of these codes. [ABSTRACT FROM AUTHOR]

A $d$ -multiplicative secret sharing ($d$ -MSS) scheme allows the players to multiply $d$ shared secrets without recovering the secrets by converting their shares locally into an additive sharing of the product. It has been proved that the $d$ -MSS among $n$ players is possible if and only if no $d$ unauthorized sets of players cover the whole set of players (type $Q_{d}$). Although this result implies some limitations on SS in the context of MPC, the $d$ -multiplicative property is still useful for simplifying complex tasks of MPC by computing the product of $d$ field elements directly and non-interactively without any setup. This paper aims to improve the usefulness of the $d$ -MSS by enhancing the security against malicious adversaries. First, we introduce the notion of verifiably multiplicative SS, verifiably MSS for short, which is mainly formalized for detecting malicious behaviors. Informally, an SS scheme is verifiably $d$ -multiplicative if the scheme is $d$ -multiplicative and further enables the players to locally generate a share of a proof that the summed value is correct (i.e., the product of $d$ shared secrets). Secondly, we prove that there is no error-free verifiably MSS scheme whose decoder of the proof is additive, and that by accepting an error probability that can be chosen arbitrarily, there exists a verifiably $d$ -MSS scheme realizing a given access structure if and only if the access structure is of type $Q_{d}$. In the proposed construction, each share of a proof consists of only two field elements. This result means that we can efficiently achieve the optimal resiliency of the standard $d$ -MSS even against malicious adversaries. We note that by allowing a general class of decoders that includes a linear one, there is an error-free verifiably $d$ -MSS scheme if the access structure is of type $Q_{d+1}$. Finally, we generalize the $d$ -multiplicative property to a $d$ -or-less version where the number $d'$ of multiplied secrets with $d'\leq d$ is not known in advance. We show that a $d$ -or-less MSS scheme can be constructed from any $d$ -MSS scheme of the same access structure with a constant overhead, and the feasibility of (verifiably) $d$ -MSS implies that of (verifiably) $d$ -or-less MSS. [ABSTRACT FROM AUTHOR]

In this paper, we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to Abelian (or multivariate) codes through the notion of ${\mathbb B}$ -apparent distance. We also study conditions for an Abelian code to verify that its ${\mathbb B}$ -apparent distance reaches its (true) minimum distance. Then, we construct some codes as an application. [ABSTRACT FROM AUTHOR]

In this paper, a new and efficient method for solving affine sub-families included in a family of nonlinear feedback shift register (NFSR) sequences is proposed. The linear case is focused on since the affine case is an analogy. Let f(x0, x1,...,xn) = x0⊕f1(x1,...,xn-1)⊕xn be a characteristic function of an n-stage NFSR, where n is a positive integer. Let deg(f) = d > 1 and f[d] be the summation of all terms in the algebraic normal form of f whose degrees attain the maximum d. First, it is proved that every linear sub-family of G(f) is a sub-family of linear feedback shift register sequences generated by a characteristic polynomial of the form Σi∈Sci xi, where ci ∈ F2 and S consists of all subscripts of variables appearing in f[d]. That is to say, every linear sub-family of G(f) is a factor of some polynomial Σi∈Sc ixi over the finite field F2. This result is a well generalization of linear recurring sequences theory since it also holds if d = 1. Based on this result, a candidate set of linear sub-families could be obtained by polynomial factorizations over F2. Second, we propose a new method to verify a linear sub-family whose memory requirement and time complexity are clearer than the previous method. For instance, all affine sub-families of the 160-bit main register used in Grain v1 could be determined within two seconds by a PC using the new method in this paper, which is unobtainable for previous algorithms. [ABSTRACT FROM AUTHOR]

Grain is one of three hardware-oriented finalists of the eSTREAM Project. A nonlinear feedback shift register (NFSR) in Grain-like structure is a cascade connection of a linear feedback shift register (LFSR) into an NFSR, in which the characteristic polynomial of the LFSR is primitive and the feedback function of the NFSR is nonsingular. In 2011 Hu and Gong pointed out that the period of the sequence generated by an NFSR in Grain-like structure is a multiple of the period of the sequence generated by its LFSR if the initial state of the LFSR is nonzero. Meanwhile, they proposed an open problem: for fixed feedback functions of an NFSR and an LFSR, determine whether the sequences generated by the NFSR in Grain-like structure can achieve the minimum period, i.e., the period of the LFSR, when the initial state of the LFSR is nonzero, and if they can achieve, provide at least one pair of the initial states of the NFSR and LFSR. Clearly, from a security point of view, it is not preferable if the sequences generated by an NFSR in Grain-like structure achieve the minimum period. This paper converts the open problem into a problem of solving an integer equation with respect to two unknown integers that uniquely correspond to the initial states of the NFSR and LFSR, by viewing the NFSR as a Boolean control network. Based on the integer equation, this paper shows that for any given initial state of an $n$ -stage NFSR and any given nonzero initial state of an $m$ -stage LFSR, the probability that the sequence generated by the NFSR in Grain-like structure achieves the minimum period $2^{m}-1$ is at most $2^{-n}$. This implies that the probability of the cascade connection used in Grain achieving the minimum period is very small. [ABSTRACT FROM AUTHOR]

This paper deals with the computational complexity of conditions which guarantee that the NP-hard problem of finding the sparsest solution to an underdetermined linear system can be solved by efficient algorithms. In the literature, several such conditions have been introduced. The most well-known ones are the mutual coherence, the restricted isometry property (RIP), and the nullspace property (NSP). While evaluating the mutual coherence of a given matrix is easy, it has been suspected for some time that evaluating RIP and NSP is computationally intractable in general. We confirm these conjectures by showing that for a given matrix \mbiA and positive integer k, computing the best constants for which the RIP or NSP hold is, in general, NP-hard. These results are based on the fact that determining the spark of a matrix is NP-hard, which is also established in this paper. Furthermore, we also give several complexity statements about problems related to the above concepts. [ABSTRACT FROM AUTHOR]

This paper studies the parameters for which binary Reed–Muller (RM) codes can be decoded successfully on the binary erasure channel and binary symmetry channel, and, in particular, when can they achieve capacity for these two classical channels. Necessarily, this paper also studies the properties of evaluations of multivariate $GF(2)$ polynomials on the random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about the square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about $E(m,r)$ , the matrix whose rows are the truth tables of all the monomials of degree $\leq r$ in $m$ variables. What is the most (resp. least) number of random columns in $E(m,r)$ that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees $r$ , which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code $C$ of sufficiently high rate, we construct a new code $C'$ obtained by tensorizing $C$ , such that for every subset $S$ of coordinates, if $C$ can recover from erasures in $S$ , then $C'$ can recover from errors in $S$ . Specializing this to the RM codes and using our results for erasures imply our result on the unique decoding of the RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent bounds from constant degree to linear degree polynomials. [ABSTRACT FROM PUBLISHER]

In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. In this paper, we slightly modify this notion to obtain the so-called irreducible-expansion complexity which is more suitable for certain applications. We analyze both, the classical and the modified expansion complexity. Moreover, we also study the expansion complexity of the explicit inversive congruential generator. [ABSTRACT FROM AUTHOR]

In pliable index coding, we consider a server with m messages and n clients, where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most \mathcal O(\log ^2(n)) broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case \mathcal O(n) transmissions. In this paper, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by \mathcal O(\log ^2(n)) broadcast transmissions. We extend our algorithm to the t -requests case, where each client requires t unknown messages that she does not have, and show that our algorithm requires at most \mathcal O(t\log (n)+\log ^2(n)) broadcast transmissions. We construct lower bound instances that require at least \Omega (\log (n)) transmissions for linear pliable index coding and at least $\Omega (t+\log (n))$ transmissions for the t$ -requests case, indicating that both our upper and lower bounds are polynomials of \log (n)$ and differ within a factor of . We provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely $\Theta (\log (n))$ , as compared with the $\Theta (n/\log (n))$ for index coding. In addition, we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances, implying that vector coding does not provide benefits in terms of these bounds. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50% less transmissions. [ABSTRACT FROM AUTHOR]

In this paper, an explicit expression of the weak-locator polynomial for $p$ -ary quadratic residue codes is presented by a modification of the Feng–Tzeng matrix method. The differences between the modified version and the original Feng–Tzeng matrix are that in the new matrix, not every entry is a syndrome, and every syndrome entry is a known syndrome. By utilizing this technique, an algebraic decoding of the ternary (61, 30, 12) quadratic residue code is proposed. This new result has never been seen in the literature to our knowledge. An advantage of the proposed decoding algorithm is that in general the obtained weak-locator polynomials can decode efficiently not only all the error patterns of weights four and five, but also some error patterns of weight six. [ABSTRACT FROM PUBLISHER]

This paper investigates the problem of characterizing the capacity of Half-Duplex (HD) line networks, where a source node communicates to a destination node through a multi-hop path of $N$ relays. If the relays operate in Full-Duplex (FD), it is well known that the capacity of the line network equals the minimum among the point-to-point link capacities in the path. In contrast, this paper considers a different case where the relays operate in HD. In the first part of the paper, it is shown that the approximate capacity (optimal up to a constant additive gap that only depends on the number of nodes in the network) of an HD $N$ -relay line network equals half the minimum of the harmonic means of the point-to-point link capacities of each two consecutive links in the path. It is then proved that the $N+1$ listen/transmit states (out of the $2^{N}$ possible ones) sufficient to characterize the approximate capacity can be found in linear time. In the second part of the paper, it is shown that the problem of finding the path that has the largest HD approximate capacity in a network that can be represented as a graph is NP-hard. However, if the number of cycles in the network is polynomial in the number of nodes, then a polynomial-time algorithm can indeed be designed. [ABSTRACT FROM AUTHOR]

Let $\mathbb {F}_{q}$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb {F}_{q} [x]$ of degree $k+m$ and a non-negative integer $r$ , count the number of polynomials $g(x)\in \mathbb {F}_{q} [x]$ of degree at most $k-1$ such that $f(x)+g(x)$ has exactly $r$ roots in $\mathbb {F}_{q}$. Previously, explicit formulas were known only for the cases $m=0, 1, 2$. As an application, we obtain an asymptotic formula on the list size of the standard Reed-Solomon code $[q, k, q-k+1]_{q}$. [ABSTRACT FROM AUTHOR]

In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input-multiple-output (MIMO) channels with constant coefficients for any number of users, antennas, and streams per user, and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove a result that generalizes those previously published on this topic. In particular, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians), and the solution set (channels, decoders, and precoders satisfying the IA polynomial equations), not only as algebraic sets, but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial complexity class. [ABSTRACT FROM AUTHOR]

Reed–Muller (RM) codes encode an m -variate polynomial of degree at most r by evaluating it on all points in \0,1\^m . We denote this code by RM(r,m) . The minimum distance of RM(r,m) is 2^{m-r} and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length n=2^{m} ) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree r=o(\sqrt m) ), we can correct a random set of (1/2-o(1))n errors with high probability. For high rate codes (of degree m-r for r=o(\sqrt {m/\log m}) ), we can correct roughly m^{r/2}$ errors. More generally, for any integer $r$ , our algorithm can correct any error pattern in $RM(m-(2r+2),m)$ , for which the same erasure pattern can be corrected in $RM(m-(r+1),m)$ . The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kötter that generalizes the Berlekamp–Welch algorithm for decoding Reed–Solomon codes. [ABSTRACT FROM PUBLISHER]

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature. [ABSTRACT FROM PUBLISHER]

This paper considers the optimum single cell power control maximizing the aggregate (uplink) communication rate of the cell when there are peak power constraints at mobile users, and a low-complexity data decoder (without successive decoding) at the base station. It is shown that the optimum power allocation is binary, which means that links are either “on” or “off.” By exploiting further structure of the optimum binary power allocation, a simple polynomial-time algorithm for finding the optimum transmission power allocation is proposed, together with a reduced complexity near-optimal heuristic algorithm. Sufficient conditions under which channel-state aware time division multiple access (TDMA) maximizes the aggregate communication rate are established. In a numerical study, we compare and contrast the performance achieved by the optimum binary power-control policy with other suboptimum policies and the throughput capacity achievable via successive decoding. It is observed that two dominant modes of communication arise, wideband or TDMA, and that successive decoding achieves better sum-rates only under near perfect interference cancellation efficiency. In this paper, we exploit the theory of majorization to obtain the aforementioned results. In the final part of this paper, we do so to solve power-control problems in the areas of femtocells and cognitive radio and find that, again, optimal solutions have a binary (or almost binary) character. [ABSTRACT FROM PUBLISHER]

Space-time block code designs based on algebraic field extension for full rate, large diversity product, and nonvanishing minimum determinant of codewords have received great attention. There are many different types of codes available for two-transmitter antennas, such as cyclotomic space-time block codes, the golden space-time block code, and rotation-based space-time block codes. In this paper, a more general space-time block code design scheme, which is called quadratic space-time block coding, is proposed for the two-transmitter antennas using quadratic field extension. The optimal design of the quadratic space-time block codes in terms of a diversity product criterion is also presented. It is shown that the optimal quadratic space-time block codes designed in this paper do not belong to the existing space-time block code family such as the cyclotomic, golden, and rotation-based space-time block codes. The simulation results demonstrate that the average codeword error rate of the optimal quadratic space-time block code attains about 0.5 dB signal to noise ratio gain over those of the optimal cyclotomic and golden space-time block codes. [ABSTRACT FROM PUBLISHER]

Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, three types of generalized cyclotomy of order two are described and three classes of cyclic codes of length n1n2 and dimension (n1n2+1)/2 are presented and analyzed, where n1 and n2 are two distinct primes. Bounds on their minimum odd-like weight are also proved. Some of the codes presented in this paper are among the best cyclic codes. [ABSTRACT FROM AUTHOR]

The $q^{t} \times (q+1)t$ ordered orthogonal arrays (OOAs) of strength $t$ over the alphabet $ \mathbb {F}_{q}$ were constructed using linear feedback shift register sequences (LFSRs) defined by primitive polynomials in $ \mathbb {F}_{q}[x]$. In this paper, we extend this result to all polynomials in $\mathbb {F}_{q}[x]$ which satisfy some fairly simple restrictions, i.e., the restrictions that are automatically satisfied by primitive polynomials. While these restrictions sometimes reduce the number of columns produced from $(q+1)t$ to a smaller multiple of $t$ , in many cases, we still obtain the maximum number of columns in the constructed OOA when using non-primitive polynomials. For $2 \le q \le 9$ and small $t$ , we generate OOAs in this manner for all permissible polynomials of degree $t$ in $ \mathbb {F}_{q}[x]$ and compare the results to the ones produced in , , and showing how close the arrays are to being “full” orthogonal arrays. Unusually for the finite fields, our arrays based on the non-primitive irreducible and even reducible polynomials are closer to the orthogonal arrays than those built from the primitive polynomials. [ABSTRACT FROM AUTHOR]

Partial syndrome matrices and weak-locator polynomials have received considerable attention in recent years due to their applications in decoding cyclic codes. The technical contribution of this paper is threefold: 1) cyclic codes can be decoded by the newly proposed partial syndrome matrices, which generalize the previously known results on the determination of error positions; 2) a new type of polynomial associated with error locations, called radical-locator polynomial, is defined, which includes the weak-locator polynomial as a special case; and 3) a novel class of matrices with many zero entries, called row-echelon partial syndrome matrices, is presented, based on the Newton identities, for efficiently decoding cyclic codes. It is also shown that the radical-locator polynomials can be obtained from the determinants of the above-mentioned (row-echelon) partial syndrome matrices. [ABSTRACT FROM AUTHOR]

In this paper, we construct some class of explicit subcovers of the curve $\mathcal {X}_{n,r}$ defined over $\mathbb {F}_{q^{n}}$ by affine equation $y^{q^{n-1}}+\cdots +y^{q}+y=x^{q^{n-r}+1}-x^{q^{n}+q^{n-r}}$. These subcovers are defined over $\mathbb {F}_{q^{n}}$ by affine equation $g_{s}(y)=x^{q^{n}+q^{n-r}}-x^{q^{n-r}+1}$ , where $g_{s}(y)$ is a $q$ -polynomial of degree $q^{s}$. The Weierstrass semigroup $H(P_\infty)$ , where $P_\infty $ is the only point at infinity on such subcovers, is determined for $1 \leq s \leq 2r-n+1$ , and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained. [ABSTRACT FROM AUTHOR]

The square $C^{*2}$ of a linear error correcting code $C$ is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in $C$. Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications, one is concerned about some of the parameters (dimension and minimum distance) of both $C^{*2}$ and $C$. In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained, and constructions of cyclic codes $C$ with a relatively large dimension of $C$ and minimum distance of the square $C^{*2}$ are discussed. In some cases, the constructions lead to codes $C$ such that both $C$ and $C^{*2}$ simultaneously have the largest possible minimum distances for their length and dimensions. [ABSTRACT FROM AUTHOR]

Signal-to-noise (SNR) ratio is commonly regarded as a reliable performance measure for wireless communication systems. Knowledge of fundamental properties of the feasible SNR region can facilitate the performance optimization of multi-user wireless systems. This paper examines the concavity of the feasible SNR region. In particular, it is shown that for systems with only three users, the feasible SNR region is always concave. As concavity for 2-D systems is well known and concavity for 4-D systems does not hold in general, this result fills in a gap on this issue. A concavity result for systems with a general number of users is also established under certain technical conditions. [ABSTRACT FROM AUTHOR]

The polarization process of polar codes over a prime $q$ -ary alphabet is studied. Recently, it has been shown that the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this paper, a different approach to computing the degree of this polynomial for any prime alphabet size is shown. This approach yields a lower degree polynomial for various values of the alphabet size that were examined. It is also shown that even lower degree polynomial can be computed with an additional numerical effort. [ABSTRACT FROM AUTHOR]

When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity. While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the number of nodes participating in the repair process (locality), at the cost of reduced minimum distance. In this paper, we study the tradeoff between locality and minimum distance of LRCs with local codes that have arbitrary distance requirements. Unlike existing methods, where both the locality and the local distance requirements imposed on every node are identical, we allow the requirements to vary arbitrarily from node to node. Such a property can be an advantage for distributed storage systems with non-homogeneous characteristics. We present Singleton-type distance upper bounds and also provide an optimal code construction with respect to these bounds. In addition, the feasible rate region is characterized by dimension upper bounds that do not depend on the distance. [ABSTRACT FROM AUTHOR]

The main problem of constant-dimension subspace coding is to determine the maximal possible size ${\mathbf{A}}_{q}(n,d,k)$ of a set of $k$ -dimensional subspaces in ${\mathbf{F}}_{q}^{n}$ such that the subspace distance satisfies $d(U,V) \geq d$ for any two different subspaces $U$ and $V$ in this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots of $L_{1}(L_{2}(x))-x$ where $L_{1}$ and $L_{2}$ are $q$ -polynomials over the extension field ${\mathbf{F}}_{q^{n}}$. New lower bounds for ${\mathbf{A}}_{q}(4k,2k,2k)$ , ${\mathbf{A}}_{q}(4k+2,2k,2k+1)$ , and ${\mathbf{A}}_{q}(4k+2,2(k-1),2k+1)$ are presented. Many new constant-dimension subspace codes better than previously best known codes with small parameters are constructed. [ABSTRACT FROM AUTHOR]

We consider Laurent polynomial (LP) matrices defined on the unit circle of the complex plane. QR decomposition of an LP matrix \bf A(s) yields QR factors \bf Q(s) and \bf R(s) that, in general, are neither LP nor rational matrices. In this paper, we present an invertible mapping that transforms \bf Q(s) and \bf R(s) into LP matrices. Furthermore, we show that, given QR factors of sufficiently many samples of \bf A(s), it is possible to obtain QR factors of additional samples of \bf A(s) through application of this mapping followed by interpolation and inversion of the mapping. The results of this paper find applications in the context of signal processing for multiple–input multiple–output (MIMO) wireless communication systems that employ orthogonal frequency-division multiplexing (OFDM). [ABSTRACT FROM PUBLISHER]

Let q = pm where p is an odd prime, m ≥ 2, and 1 ≤ k ≤ m - 1. Let Tr be the trace mapping from Fq to Fp and (ζp = e 2π i/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: Σ x ϵ Fq χ (αxpk+1 + βx²) (α, β ϵ Fq) where χ(x) = ζpTr(x) is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1 and C2 over Fpt with parity-check polynomial h2 (x)h3 (x) and h1 (x)h2 (x)h3 (x), respectively, where t is a divisor of d = gcd (m, k), and h1 (x), h2 (x), and h3 (x) are the minimal polynomials of π-1, π-2, and π-(Pk+1) over FPt respectively, for a primitive element π of Fq. 2) We determine the correlation distribution between two m-sequences of period q -1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008). [ABSTRACT FROM AUTHOR]

Recently, an algebraic decoding algorithm suggested by Truong et al. (2005) for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial. In this paper, based on the ideas of the algorithm mentioned above, an algebraic decoder for the (89, 45, 17) binary quadratic residue code, the last one not decoded yet of length less than 100, is proposed. It was also verified theoretically for all error patterns within the error-correcting capacity of the code. Moreover, the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations. [ABSTRACT FROM AUTHOR]

Gabidulin codes can be seen as the rank-metric equivalent of Reed–Solomon codes. It was recently proved, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially sized lists exist for codes with specific parameters. The second bound presents exponentially sized lists explicitly for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding several families of Gabidulin codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes. Using a standard operation called lifting, identical results also follow for an important class of constant dimension subspace codes. [ABSTRACT FROM AUTHOR]

In this paper, we discover that univariate form of a Niho bent function is a sum of functions having the form of a Leander–Kholosha bent function taken with particular coefficients from \mathbb F2^{n}^{*} for every term. We know that the Niho bent functions are related to o-polynomials. The power terms in the univariate Niho bent function can be derived by working, in a first step, on each monomial of the corresponding o-polynomial separately, and in a second step, adding them to obtain the global expression. This allows, knowing the monomials in an o-polynomial, to obtain the power terms of the polynomial representing corresponding bent function. However, the coefficients are not calculated explicitly. The explicit form is given for the bent functions obtained from quadratic and cubic o-polynomials. We also calculate the algebraic degree of any bent function in the Leander–Kholosha class. [ABSTRACT FROM AUTHOR]

In this paper, we study the data exchange problem, where a set of users is interested in gaining access to a common file, but where each has only partial knowledge about it as side-information. Assuming that the file is broken into packets, the side-information considered is in the form of linear combinations of the file packets. Given that the collective information of all the users is sufficient to allow recovery of the entire file, the goal is for each user to gain access to the file, while minimizing some communication cost. We assume that the users can communicate over a noiseless broadcast channel, and that the communication cost is a sum of each user’s cost function over the number of bits it transmits. For instance, the communication cost could simply be the total number of bits that needs to be transmitted. In the most general case studied in this paper, each user can have any arbitrary convex cost function. We provide deterministic, polynomial-time algorithms (in the number of users and packets), which find an optimal communication scheme that minimizes the communication cost. To further lower the complexity, we also propose a simple randomized algorithm inspired by our deterministic algorithm, which is based on a random linear network-coding scheme. [ABSTRACT FROM AUTHOR]

Subspace codes have received an increasing interest recently due to their application in error correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper, we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes, which are cyclic and analyze their parameters. [ABSTRACT FROM AUTHOR]

For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized Reed–Solomon (GRS) codes is proved to be co-NP-complete by Guruswami and Vardy. For the extended Reed–Solomon codes RSq( \mathbb {F}q,k) , a conjecture was made to classify deep holes by Cheng and Murray. Since then efforts have been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for GRS codes RSp (D,k) , where $p$ is a prime, |D| > k \geqslant ({p-1})/{2} . Our techniques are built on the idea of deep hole trees, and several results concerning the Erdös–Heilbronn conjecture. [ABSTRACT FROM PUBLISHER]