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An $[{n}, {k}, {n}-{k}+1]$ linear code is called an MDS code. An $[{n}, {k}, {n}-{k}]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system ${S}(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for $ {t}\geq 2$ remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over $ {GF}(3^{ {s}})$ holding an infinite family of 3-designs and an infinite family of near MDS codes over $ {GF}(2^{2 {s}})$ holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal. [ABSTRACT FROM AUTHOR]

Quasi-cyclic (QC) low-density parity-check (LDPC) codes which are known as QC-LDPC codes, have many applications due to their simple encoding implementation by the means of cyclic shift registers. In this paper, we construct QC-LDPC codes from group rings. A group ring is a free module (at the same time a ring) constructed in a natural way from any given ring and any given group. We present a structure based on the elements of a group ring for constructing QC-LDPC codes. Some of the previously addressed methods for constructing QC-LDPC codes based on finite fields are special cases of the proposed construction method. The constructed QC-LDPC codes perform very well over the additive white Gaussian noise channel with iterative decoding in terms of bit-error probability and block-error probability. Simulation results demonstrate that the proposed codes have competitive performance in comparison with the similar existing LDPC codes. Finally, we propose a new encoding method for the proposed group ring-based QC-LDPC codes that can be implemented faster than the current encoding methods. The encoding complexity of the proposed method is analyzed mathematically, and indicates a significate reduction in the required number of operations, even when compared to the available efficient encoding methods that have linear time and space complexities. [ABSTRACT FROM PUBLISHER]

The Rosenbloom–Tsfasman metric has attracted the attention of many researchers as a generalization of the Hamming metric that is relevant to practical problems. Codes for this metric were considered. In particular, Reed–Solomon codes were generalized to be compatible with this metric. In this paper, a generalization of BCH codes for the Rosenbloom–Tsfasman metric is proposed. This generalization is based on considering BCH codes as subfield subcodes of Reed–Solomon codes. By characterizing these subfield subcodes, an explicit construction of BCH codes for the Rosenbloom–Tsfasman metric is provided. Two important properties of Reed–Solomon codes and BCH codes for the Rosenbloom–Tsfasman metric are studied and compared with those for the Hamming metric. These properties are cyclic structure and duality. The approach is based on Galois-Fourier transforms associated with Hasse derivatives. [ABSTRACT FROM PUBLISHER]

In this paper, we consider optimal q -ary cyclic constant-weight codes of length n , minimum distance d , and weight w codes. We introduce the pure and mixed difference method to present a combinatorial description for a cyclic (n,d,w)_{q} code and then obtain some tight upper bounds on the sizes of optimal cyclic (n,d,w)_{q} codes. Finally, by using Skolem-type sequences, we completely determine the sizes of optimal cyclic (n,d,3)3 codes with minimum distance 1\leq d\leq 6 . [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$ , i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this paper, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$ , dimension $k$ , locality $r$ , availability $t$ , and field size $q$ . Then, we study the binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed–Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared with other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, the availability of LRCs is studied. We investigate the locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check codes. We also show the construction of a long LRC with availability from a short one-step majority-logic decodable code. [ABSTRACT FROM PUBLISHER]

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, the two-prime sequence is employed to construct several classes of cyclic codes over GF(q). Lower bounds on the minimum weight of these cyclic codes are developed. Some of the codes obtained are optimal or almost optimal. The p-ranks of the twin-prime difference sets and a class of almost difference sets are computed. [ABSTRACT FROM PUBLISHER]

Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases. In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few more cases. [ABSTRACT FROM AUTHOR]

In communication systems, frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are used in FH-CDMA systems. In this paper, an earlier idea of constructing optimal sets of frequency hopping sequences is further investigated. New optimal parameters of sets of frequency hopping sequences are obtained with subcodes of the Reed-Solomon codes. Optimal sets of frequency hopping sequences are constructed with a class of irreducible cyclic codes. As a byproduct, the weight distribution of a subclass of irreducible cyclic codes is determined. [ABSTRACT FROM AUTHOR]

Let $p$ be a prime, $s$ be a positive integer, and let $b$ be an integer satisfying $2 \leq b < p^{s}$. In this paper, we obtain $b$ -symbol distances of all repeated-root constacyclic codes of length $p^{s}$ over finite fields. Using this result, we determine $b$ -symbol distances of all repeated-root constacyclic codes of length $p^{s}$ over finite commutative chain rings. We also list all MDS $b$ -symbol repeated-root constacyclic codes of length $p^{s}$ over finite fields, and all MDS $b$ -symbol repeated-root constacyclic codes of length $p^{s}$ over finite commutative chain rings in general. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros. [ABSTRACT FROM PUBLISHER]

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes. [ABSTRACT FROM AUTHOR]

Symbol-pair read channels, in which the outputs of the read process are pairs of consecutive symbols, were recently studied by Cassuto and Blaum. This new paradigm is motivated by the limitations of the reading process in high density data storage systems. They studied error correction in this new paradigm, specifically, the relationship between the minimum Hamming distance of an error correcting code and the minimum pair distance, which is the minimum Hamming distance between symbol-pair vectors derived from codewords of the code. It was proved that for a linear cyclic code with minimum Hamming distance dH , the corresponding minimum pair distance is at least dH+3 . In this paper, we show that, for a given linear cyclic code with a minimum Hamming distance dH , the minimum pair distance is at least dH +\left \lceil{ {dH/{2}}}\right \rceil . We then describe a decoding algorithm, based upon a bounded distance decoder for the cyclic code, whose symbol-pair error correcting capabilities reflect the larger minimum pair distance. Finally, we consider the case where the read channel output is a larger number, $b \geqslant 3$ , of consecutive symbols, and we provide extensions of several concepts, results, and code constructions to this setting. [ABSTRACT FROM AUTHOR]

A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb {Z}}_{2}$ coordinates and the set of ${\mathbb {Z}}_{4}$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let $\Phi ({\mathcal{ C}})$ be the binary Gray map image of ${\mathcal{ C}}$. We study the rank and the dimension of the kernel of a ${\mathbb {Z}}_{2} {\mathbb {Z}}_{4} $ -additive cyclic code ${\mathcal{ C}}$ , that is, the dimensions of the binary linear codes $\langle \Phi ({\mathcal{ C}}) \rangle $ and $\ker (\Phi ({\mathcal{ C}}))$. We give upper and lower bounds for these parameters. It is known that the codes $\langle \Phi ({\mathcal{ C}}) \rangle $ and $\ker (\Phi ({\mathcal{ C}}))$ are binary images of ${\mathbb {Z}}_{2} {\mathbb {Z}}_{4}$ -additive codes that we denote by ${\mathcal{ R}}({\mathcal{ C}})$ and ${\mathcal{ K}}({\mathcal{ C}})$ , respectively. Moreover, we show that ${\mathcal{ R}}({\mathcal{ C}})$ and ${\mathcal{ K}}({\mathcal{ C}})$ are also cyclic and determine the generator polynomials of these codes in terms of the generator polynomials of the code ${\mathcal{ C}}$. [ABSTRACT FROM AUTHOR]

Quadratic residue codes are a well-known class of codes. In this paper, we consider the constructions of self-dual codes by higher power residues, especially fourth power residues. New infinite families of self-dual codes over GF(2), GF(3), GF(4), GF(8), and GF(9) are introduced. Some of them have better minimum weight than previously known codes. We also give general results related to the automorphism group of some of these codes. [ABSTRACT FROM PUBLISHER]

Let \(m\geq 3\) be an odd integer and \(p\) be an odd prime. In this paper, a number of classes of three-weight cyclic codes \(\mathcal {C}_{(1,e)}\) over \(\mathbb {F}_{p}\) , which have parity-check polynomial \(m_1(x)m_e(x)\) , are presented by examining general conditions on the parameters \(p\) , \(m\) , and \(e\) , where \(m_i(x)\) is the minimal polynomial of \(\pi ^{-i}\) over \(\mathbb {F}_{p}\) for a primitive element \(\pi \) of \(\mathbb {F}_{p^m}\) . Furthermore, for \(p\equiv 3 \pmod {4}\) and a positive integer \(e\) satisfying \((p^k+1)\cdot e\equiv 2 \pmod {p^m-1}\) for some positive integer \(k\) with \(\gcd (m,k)=1\) , the value distributions of the exponential sums \(T(a,b)=\sum \nolimits _{x\in \mathbb {F}_{p^m}}\omega ^{ \text {Tr}(ax+bx^e)}\) and \(S(a,b,c)=\sum \nolimits _{x\in \mathbb {F}_{p^m}}\omega ^{ \text {Tr}(ax+bx^e+cx^s)},\) where \(s=(p^m-1)/2\) , are determined. As an application, the value distribution of \(S(a,b,c)\) is utilized to derive the weight distribution of the cyclic codes \(\mathcal {C}_{(1,e,s)}\) with parity-check polynomial \(m_1(x)m_e(x)m_s(x)\) . In the case of \(p=3\) and even \(e\) satisfying the above condition, the dual of the cyclic code \(\mathcal {C}_{(1,e,s)}\) has optimal minimum distance. [ABSTRACT FROM PUBLISHER]

Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents. More specifically, we obtain two classes of binary three-weight and four-weight cyclic codes and a class of nonbinary four-weight cyclic codes. The weight distributions follow from the determination of value distributions of certain exponential sums. Several examples are presented to show that some of our codes are optimal and some have the best known parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study \BBZ2\BBZ4-additive cyclic codes. These codes are identified as \BBZ4[x]-submodules of the ring Rr,s=\BBZ2[x]/\langle x^{r}-1\rangle\times\BBZ4\left[x\right]/\langle x^{s}-1\rangle. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a \BBZ4[x]-submodule of the ring Rr,s is determined. We show that the duals of \BBZ2\BBZ4-additive cyclic codes are also cyclic. We also present an infinite family of Maximum Distance separable with respect to the singleton bound codes. Finally, we obtain a number of binary linear codes with optimal parameters from the \BBZ2\BBZ4-additive cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the cycle properties of quasi-cyclic low-density parity-check (QC-LDPC) codes. Using the sequence representation of a parity-check matrix for a QC-LDPC code, we analyze a necessary and sufficient condition for a cycle of a given length to exist. We then derive bounds which are necessary conditions for a QC-LDPC code to have a given girth in terms of its parameters. We also give a bound which is a sufficient condition for a QC-LDPC code of a given girth to be constructed by a greedy algorithm. The bounds derived here are applicable to any regular or irregular QC-LDPC codes as well as they improve the existing bounds in many classes of regular LDPC codes. [ABSTRACT FROM AUTHOR]

In this paper, we present an analog of the Singleton bound for convolutional codes over finite rings. Codes meeting that bound are called MDS. A constructive method for the MDS codes, restricted to free codes, is presented via cyclic codes over finite rings. Examples of nonfree MDS codes are provided. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of p-ary cyclic codes whose duals have three pairwise nonconjugate zeros is proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights. [ABSTRACT FROM AUTHOR]

Cyclic codes have been widely used in digital communication systems and consumer electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros is described and their weight distributions are determined. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p^2m-1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of p-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even m and odd m along with the idea of Li The weight distributions of two related families of cyclic codes are also determined. [ABSTRACT FROM AUTHOR]

The duals of cyclic codes with two zeros have been extensively studied, and their weight distributions have recently been evaluated in some cases. In this paper, we determine the weight distribution of a certain new class of such codes by computations involving index 2 Gauss sums. [ABSTRACT FROM AUTHOR]

The projective special linear group ${\mathrm {PSL}}_{2}(n)$ is 2-transitive for all primes $n$ and 3-homogeneous for $n \equiv 3 \pmod {4}$ on the set $\{0,1, \ldots, n-1, \infty \}$. It is known that the extended odd-like quadratic residue codes are invariant under ${\mathrm {PSL}}_{2}(n)$. Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes $n \equiv 1 \pmod {4}$ , an infinite family of 3-designs for primes $n \equiv 3 \pmod {4}$. To construct more $t$ -designs with $t \in \{2, 3\}$ , one would search for other extended cyclic codes over finite fields that are invariant under the action of ${\mathrm {PSL}}_{2}(n)$. The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under ${\mathrm {PSL}}_{2}(n)$. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new family of quantum synchronizable codes from repeated-root cyclic codes of lengths p^s and lp^s over \mathbb Fq , where $s\geq 1~and l\geq 2$ are integers, and $p\geq 3$ is the odd characteristic. Within some loose limitations, these synchronizable codes can possess the best possible capability in synchronization recovery, and therefore, enriches the variety of good quantum synchronizable codes. Furthermore, by using known techniques in classical coding theory which convert the computation of the minimum distance of a repeated-root cyclic code to that of a shorter simple-root cyclic code, we prove that the repeated-root cyclic codes of lengths p^s and lp^s are in general better than narrow-sense BCH codes of close lengths in terms of minimum distances, and thereby enable the obtained synchronizable codes to correct more Pauli errors. [ABSTRACT FROM PUBLISHER]

Linear network coding over finite fields is a well-studied problem. We consider the more general setting of linear coding for directed acyclic networks with finite commutative ring alphabets. Our results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but other rings may be needed when the alphabet size is fixed. We prove that if a network has a scalar linear solution over some finite commutative ring, then the (unique) smallest such commutative ring is a field. We also show that fixed-size commutative rings are quasi-ordered, such that all the scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. We study commutative rings that are maximal with respect to this quasi-order, as they may be considered the best commutative rings of a given size. We prove that a commutative ring is maximal if and only if some network is scalar linearly solvable over the ring, but not over any other commutative ring of the same size. Furthermore, we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the ring’s size. Finally, we prove that there is a unique maximal commutative ring of size $m$ if and only if each prime factor of m$ has multiplicity in {1, 2, 3, 4, 6}. As consequences, 1) every finite field is such a maximal ring and 2) for each prime p$ , some network is scalar linearly solvable over a commutative ring of size but not over the field of the same size if and only if k \not \in \{1,2,3,4,6\} . [ABSTRACT FROM PUBLISHER]

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First, we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second, we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes. [ABSTRACT FROM AUTHOR]

Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes have been studied for many years and find many applications. The determination of the weight distributions of such codes is an open problem. Generally, the weight distributions of cyclic codes are difficult to determine. Utilizing a class of elliptic curves, this paper determines the weight distributions of dual codes of q-ary cyclic codes with two zeros for a few more cases, where q is an odd prime power. [ABSTRACT FROM AUTHOR]

Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases. In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few cases. The weight distributions of punctured versions of these codes are also determined for several special cases. [ABSTRACT FROM AUTHOR]

Cyclic linear codes of block length n over a finite field Fq are linear subspaces Fqn of that are invariant under a cyclic shift of their coordinates. A family of codes is good if all the codes in the family have constant rate and constant normalized distance (distance divided by block length). It is a long-standing open problem whether there exists a good family of cyclic linear codes. A code C is r-testable if there exists a randomized algorithm which, given a word x ∈ Fqn, adaptively selects r positions, checks the entries of x in the selected positions, and makes a decision (accept or reject x) based on the positions selects and the numbers found, such that i) jf x ∈ C then x is surely accepted; ii) if dist (x, C) ⩾ ∈n then x is probably rejected. ("dist" refers to Hamming distance.) A family of codes is locally testable if all members of the family are r-testable for some constant r. This concept arose from holographic proofs/PCP's. Recently it was asked whether there exist good, locally testable families of codes. In this paper the intersection of the two questions stated is addressed. Theorem. There are no good, locally testable families of cyclic codes over any (fixed) finite field. In fact the result is stronger in that it replaces condition ii) of local testability by the condition ii) if dist (x, C) ⩾ en then x has a positive chance of being rejected. The proof involves methods from Galois theory, cyclotomy, and diophantine approximation. [ABSTRACT FROM AUTHOR]

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega’s result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights dr(C) of a linear code C are a natural generalization of the minimum Hamming distance d(C)[=d_{1}(C)] and have become an important research object in coding theory since Wei’s originary work in 1991. In this paper, two general formulas on dr(C) for irreducible cyclic codes are presented using Gauss sums and the weight hierarchy \{d_{1}(C), d_{2}(C), \ldots , d_{k}(C)\}~(k=\dim C)$ is completely determined for several cases. [ABSTRACT FROM AUTHOR]

This corrects a proposition and its proof in the paper “Sharp p-Divisibility of Weights in Abelian Codes over \BBZ/p^d\BBZ” by Daniel J. Katz (2008). This proposition was used to prove further results in the paper. The rest of the paper (including the main results and all other claims, and their proofs) stands as written, for it is not at all sensitive to the change. [ABSTRACT FROM PUBLISHER]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let \(\Bbb F_r\) be a finite field with \(r=q^m\) . Suppose that \(g_1, g_2 \in \Bbb F_r^*\) are not conjugates over \(\Bbb F_q\) , \( \operatorname {ord}(g_1)=n_1\) , \( \operatorname {ord}(g_2)=n_2\) , \(d=\gcd (n_1, n_2)\) , and \(n={n_1n_2}/ d\) . Let \(\Bbb F_q(g_1)=\Bbb F_{q^{m_1}}, \Bbb F_q(g_2)=\Bbb F_{q^{m_2}}\) , and \(\text{T}_i\) denote the trace function from \(\Bbb F_{q^{m_i}}\) to \(\Bbb F_q\) for \(i=1, 2\) . We define a cyclic code \(\mathcal C_{(q, m, n_1, n_2)}=\{c(a,b) : a \in \Bbb F_{q^{m_1}}, b \in \Bbb F_{q^{m_2}}\},\) where \(c(a,b)=(\text {T}_1(ag_1^0)+\text {T}_2(bg_2^0), \text {T}_1(ag_1^1)+\text {T}_2(bg_2^1), \ldots , \text {T}_1(ag_1^{n-1})+\text {T}_2(bg_2^{n-1})).\) We mainly use Gauss periods to present the weight distribution of the cyclic code \(\mathcal C_{(q, m, n_1, n_2)}\) . As applications, we determine the weight distribution of cyclic code \(\mathcal C_{(q, m,q^{m_1}-1,q^{m_2}-1)}\) with \(\gcd (m_1,m_2)=1\) ; in particular, it is a three-weight cyclic code if \(\gcd (q-1,m_1-m_2)=1\) . We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight distributions are known only for a few cases. In this paper, let \BBFr be an extension of a finite field \BBFq and r=q^m, we determine the weight distributions of the cyclic codes \cal C=\c(a, b): a, b\in\BBFr\, c(a, b)=(Trr/q(ag1^{0}+bg2^{0}),\ldots,Trr/q(ag1^{n-1}+bg2^{n-1})), g1, g2\in\BBFr, in the following two cases: 1) \mathopord(g1)=n, n\vert r-1 and g2=1 and 2) \mathopord(g1)=n, g2=g1^{2}, \mathopord(g2)=n/2, m=2, and 2(r-1)/n\vert (q+1). [ABSTRACT FROM PUBLISHER]

In this paper, the minimum weight and the dimension of all cyclic codes of length p^n over a field \BBFq, are computed, when p is an odd prime and \BBFq a finite field with q elements, assuming that \overlineq generates the group of invertible elements of \BBZp^{n}. Furthermore, the minimum weight and dimension of codes which are sum of two minimal codes in \BBFq(Cp\times Cp) are also computed. Finally, the efficiency of cyclic codes and noncyclic abelian codes of length p^2 are compared. [ABSTRACT FROM AUTHOR]

We introduce a new class of rate one-half binary codes: complementary information set codes. A binary linear code of length 2n and dimension n is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune vectorial Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper, we investigate this new class of codes: we give optimal or best known CIS codes of length < 132. We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov–Gilbert bound for long CIS codes, and show that they can all be classified in small lengths \leq 12 by the building up construction. Some nonlinear permutations are constructed by using \BBZ4-codes, based on the notion of dual distance of a possibly nonlinear code. [ABSTRACT FROM AUTHOR]

We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs), where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed–Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al.. Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol. For all three protocols, we present an algorithm to optimize their PIR rates. [ABSTRACT FROM AUTHOR]

A new coding framework is established for channels whose outputs are overlapping pairs of symbols. Such channels are motivated by storage applications in which the spatial resolution of the reader may be insufficient to isolate adjacent symbols. Reading symbols as pairs changes the coding-theoretic error model from the standard bounded number of symbol errors to a bounded number of pair errors. Starting from the most basic coding-theoretic questions, the paper studies codes that protect against pair-errors. It provides answers on pair-error correctability conditions, code construction and decoding, and lower and upper bounds on code sizes. Asymptotic analysis of pair-error correction shows that there exist pair-error codes with rates that are strictly higher than the best known codes in the Hamming metric. [ABSTRACT FROM AUTHOR]

The guessing number of a directed graph (digraph), equivalent to the entropy of that digraph, was introduced as a direct criterion on the solvability of a network coding instance. This paper makes two contributions on the guessing number. First, we introduce an undirected graph on all possible configurations of the digraph, referred to as the guessing graph, which encapsulates the essence of dependence amongst configurations. We prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph. Therefore, network coding solvability is no more a problem on the operations made by each node, but is simplified into a problem on the messages that can transit through the network. By studying the guessing graph of a given digraph, and how to combine digraphs or alphabets, we are thus able to derive bounds on the guessing number of digraphs. Second, we construct specific digraphs with high guessing numbers, yielding network coding instances where a large amount of information can transit. We first propose a construction of digraphs with finite parameters based on cyclic codes, with guessing number equal to the degree of the generator polynomial. We then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one, despite these digraphs being arbitrarily sparse. These constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear, although these instances are sparse and the sources are arbitrarily far from their corresponding sinks. [ABSTRACT FROM AUTHOR]

A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb {Z}}_{2}$ and the set of ${\mathbb {Z}}_{4}$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of $\mathbb {Z}_{2} \mathbb {Z}_{4} $ -additive cyclic codes. We determine all ${\mathbb {Z}_{2}\mathbb {Z}_{4}}$ -additive cyclic codes with odd $\beta $ whose Gray images are linear binary codes. In this case, it is shown that such binary codes are permutation equivalent (by the Nechaev permutation) to $\mathbb {Z}_{2}$ -double cyclic codes. Finally, the generator polynomials of these binary codes are given. [ABSTRACT FROM AUTHOR]

We introduce the notion of weakly mutually uncorrelated (WMU) sequences, motivated by applications in DNA-based data storage systems and synchronization between communication devices. WMU sequences are characterized by the property that no sufficiently long suffix of one sequence is the prefix of the same or another sequence. WMU sequences used for primer design in DNA-based data storage systems are also required to be at large mutual Hamming distance from each other, have balanced compositions of symbols, and avoid primer-dimer byproducts. We derive bounds on the size of WMU and various constrained WMU codes and present a number of constructions for balanced, error-correcting, primer-dimer free WMU codes using Dyck paths, prefix-synchronized, and cyclic codes. [ABSTRACT FROM AUTHOR]

Three new families of lowest density maximum-distance separable (MDS) array codes are constructed, which are cyclic or quasi-cyclic. In addition to their optimal redundancy (MDS) and optimal update complexity (lowest density), the symmetry offered by the new codes can be utilized for simplified implementation in storage applications. The proof of the code properties has an indirect structure: first MDS codes that are not cyclic are constructed, and then transformed to cyclic codes by a minimum-distance preserving transformation. [ABSTRACT FROM AUTHOR]

A theorem of McEliece on the p-divisibility of Hamming weights in cyclic codes over Fp is generalized to Abelian codes over Z/pd Z. This work improves upon results of Helleseth—Kumar—Moreno—Shanbhag, Calderbank—Li—Poonen, Wilson, and Katz. These previous attempts are not sharp in general, i.e., do not report the full extent of the p-divisibility except in special cases, nor do they give accounts of the precise circumstances under which they do provide best possible results. This paper provides sharp results on p-divisibilities of Hamming weights and counts of any particular symbol for an arbitrary Abelian code over Z/pd Z. It also presents sharp results on 2-divisibilities of Lee and Euclidean weights for Abelian codes over Z/4Z. [ABSTRACT FROM AUTHOR]

In this paper, we show that a recently proposed algorithm for decoding cyclic codes may be applied efficiently to all binary cyclic codes with t ⩽ 2 and n < 63. This is accomplished by providing structure theorems for the codes in this range and classifying the relevant cases. [ABSTRACT FROM AUTHOR]

We present algorithms for initializing a convolutional network coding (CNC) scheme in networks that may contain cycles. An initialization process for finding global encoding kernels (GEK) is needed if the network is unknown or if local encoding kernels are chosen randomly. During the initialization process every source node transmits basis vectors and every sink node gets the impulse response of the network. The impulse response is then used to find the GEK, which are needed for a decoding algorithm and to find the set of all achievable rates. We present two initialization algorithms that find the GEK and one algorithm that finds achievable rates from the GEK. In the first initialization algorithm it is assumed that we can perform a reset operation on the network at some fixed times, while the second algorithm does not operate under this assumption. Unlike acyclic networks, for which it is sufficient to transmit basis vectors one after another, the initialization of cyclic networks is more involved, as test symbols from different times interfere with each other and the impulse response is of infinite duration. Our algorithms use only a finite number of the initial values of the impulse response to find the full GEK. This is possible because a CNC scheme can be described by a state space representation and, using the Cayley-Hamilton theorem, it is possible to find its full impulse response from its initial values. [ABSTRACT FROM AUTHOR]

For almost-cyclostationary processes, sufficient conditions are derived, such that estimates of the second-order cyclic probabilistic functions with estimated cycle frequencies are mean-square consistent and asymptotically complex normal. Cyclic (conjugate) autocorrelation functions and cyclic (conjugate) spectra are considered. Under the derived conditions, asymptotically, as the data-record length approaches infinity, the estimates of cyclic statistics with estimated cycle frequencies have the same complex normal distribution as the case of exactly known cycle frequencies. The results are applied to the detection of a moving cyclostationary source in the presence of strong Doppler effect and for low values of SNR. [ABSTRACT FROM PUBLISHER]

A \mathbb Z2\mathbb Z4 -additive code \mathcal C\subseteq \mathbb Z2^\alpha \times \mathbb Z4^\beta is called cyclic if the set of coordinates can be partitioned into two subsets, the set of \mathbb Z2 and the set of \mathbb Z4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the \mathbb Z4[x] -module \mathbb Z2[x]/(x^\alpha -1)\times \mathbb Z4[x]/(x^\beta -1) . The parameters of a \mathbb Z2\mathbb Z4 -additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a \mathbb Z2\mathbb Z4 -additive cyclic code are determined in terms of the generator polynomials of the code \mathcal C . [ABSTRACT FROM PUBLISHER]

Hermitian self-dual abelian codes in a group ring \BBFq^{2}[G], where \BBFq^{2} is a finite field of order q^2 and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semisimple group ring, a characterization of Hermitian self-dual abelian codes in \BBFq^{2}[G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in \BBFq^{2}[G], i.e., there exists a Hermitian self-dual abelian code in \BBFq^{2}[G] if and only if the order of G is even and q=2^l for some positive integer l. Later on, the study is further restricted to the case where \BBF2^{2l}[G] is a principal ideal group ring, or equivalently, G\cong A\oplus\BBZ2^{k} with 2\nmid\vert A\vert. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in \BBF{2^{2l}}[A\oplus\BBZ2^{k}] can be determined easily. When A is cyclic, this result answers an open problem of Jia concerning Hermitian self-dual cyclic codes. In many cases, \BBF{2^{2l}}[A\oplus\BBZ2^{k}] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in \BBF{2^{2l}}[A\oplus\BBZ2] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2 m over \BBF{2^{2l}} exists. [ABSTRACT FROM AUTHOR]

We give a construction of quantum LDPC codes of dimension $\Theta (\log N)$ and distance $\Theta (N/\log N)$ as the code length $N\to \infty $. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance $\Omega (N^{1-\alpha /2}/\log N)$ and dimension $\Omega (N^\alpha \log N)$ , where $0 \le \alpha < 1$. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed $R < 1$ there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least $R$ with, in some sense, optimal circulant size $\Omega (N/\log N)$ as the code length $N\to \infty $. [ABSTRACT FROM AUTHOR]