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The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper. [ABSTRACT FROM AUTHOR]

The binary quadratic-residue codes and the punctured Reed-Muller codes ${\mathcal {R}}_{2}((m-1)/2, m))$ are two families of binary cyclic codes with parameters $[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length $2^{m}-1$ and dimension near $2^{m-1}$ with good minimum distances. When $m \geq 3$ is odd, the codes become a family of duadic codes with parameters $[2^{m}-1, 2^{m-1}, d]$ , where $d \geq 2^{(m-1)/2}+1$ if $m \equiv 3 \pmod {4}$ and $d \geq 2^{(m-1)/2}+3$ if $m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes. [ABSTRACT FROM AUTHOR]

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ MDS codes for $1 \leq u \leq 2^{m-1}$ , which are cyclic, reversible and BCH codes over ${\mathrm {GF}}(2^{m})$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented. [ABSTRACT FROM AUTHOR]

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]

Given a ${q}$ -ary frequency hopping sequence set of length ${n}$ and size ${M}$ with Hamming correlation ${H}$ , one can obtain a ${q}$ -ary (nonlinear) cyclic code of length ${n}$ and size nM with Hamming distance n-H. Thus, every upper bound on the size of a code from coding theory gives an upper bound on the size of a frequency hopping sequence set. Indeed, all upper bounds from coding theory have been converted to upper bounds on frequency hopping sequence sets. On the other hand, a lower bound from coding theory does not automatically produce a lower bound for frequency hopping sequence sets. In particular, the most important lower bound, the Gilbert-Varshamov bound in coding theory, has not been transformed to a valid lower bound on frequency hopping sequence sets. The purpose of this paper is to transform the Gilbert-Varshamov bound from coding theory to frequency hopping sequence sets by establishing a connection between a special family of cyclic codes (which are called hopping cyclic codes in this paper) and frequency hopping sequence sets. We provide two proofs of the Gilbert-Varshamov bound. One is based on a probabilistic method that requires advanced tool–martingale. This proof covers the whole rate region. Another proof is purely elementary but only covers part of the rate region. [ABSTRACT FROM AUTHOR]

Motivated by the recent studies on $\mathbb {Z}_{2}\mathbb {Z}_{4}$ -additive cyclic codes, $\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$ -cyclic codes and $\mathbb {Z}_{2}\mathbb {Z}_{2^{s}}$ -additive codes have been introduced by Aydogdu et al.. In this paper, we study $\mathbb {Z}_{q}R$ -linear cyclic codes where $R=\mathbb {Z}_{q}+u\mathbb {Z}_{q}, q$ is a prime number and $u^{2}=0$. There are two major ingredients in this paper. The first is to investigate the algebraic structure of cyclic codes and their duals over ring $\mathbb {Z}_{q}R$ , the spanning sets, the types and sizes of $\mathbb {Z}_{q}R$ -cyclic codes and their duals as well. Based on this, the second ingredient is to present an infinite family of MDSS codes and obtain some illustrative examples of $q$ -ary cyclic codes with optimal parameters derived from the $\mathbb {Z}_{q}R$ -cyclic codes. [ABSTRACT FROM AUTHOR]

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length $n$ over $\mathbb {F}_{q}$ is always relative to an $n$ -th primitive root of unity $\beta $ in an extension field of $\mathbb {F}_{q}$ , and is called a dually-BCH code if its dual is also a BCH code relative to the same $\beta $. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances $\delta $ will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances $\delta $. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated. [ABSTRACT FROM AUTHOR]

Self-orthogonal codes are an interesting type of linear codes due to their wide applications in communication and cryptography. It is known that self-orthogonal codes are often used to construct quantum error-correcting codes, which can protect quantum information in quantum computations and quantum communications. Let $\mathcal C$ be an $[n, k]$ cyclic code over $\mathbb {F}_{q}$ , where $\mathbb {F}_{q}$ is the finite field of order $q$. The hull of $\mathcal C$ is defined to be the intersection of the code and its dual. In this paper, we will employ the defining sets of cyclic codes to present two general characterizations of the hulls that have dimension $k-1$ or $k^\perp -1$ , where $k^\perp $ is the dimension of the dual code $\mathcal C^\perp $. Several sufficient and necessary conditions for primitive and projective BCH codes to have $(k-1)$ -dimensional (or $(k^\perp -1)$ -dimensional) hulls are also developed by presenting lower and upper bounds on their designed distances. Furthermore, several classes of self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. The dimensions and minimum distances of some self-orthogonal codes are determined explicitly. In addition, several optimal codes are obtained. [ABSTRACT FROM AUTHOR]

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]

Locally recoverable codes are very important due to their applications in distributed storage systems. In this paper, by using cyclic codes, we construct two classes of optimal q-ary cyclic $(\text {r}, \delta _{1})$ locally recoverable codes with parameters $[2(\text {q}+1), 2(\text {q}+1)-2\delta _{1}, \delta _{1}+2]_{\text {q}}$ , where $q$ is an odd prime power and $\text {r}+\delta _{1}-1=\text {q}+1$ , and $(\text {r}, \delta _{2})$ locally recoverable codes with parameters $[2(\text {q}+1), 2(\text {q}+1)-4\delta _{2}+2, \delta _{2}+2]_{q}$ , where $\text {q}\geq 7$ is an odd prime power, $\text {q}\equiv 3~ (mod~ 4)$ and $r+\delta _{2}-1=\frac {\text {q}+1}{2}$. Compared with the known cyclic and constacyclic $(\text {r}, \delta)$ locally recoverable codes, our construction yields new optimal cyclic $(\text {r}, \delta)$ locally recoverable codes. [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]

Fire codes are cyclic codes generated by the product of two polynomials: a binomial that characterizes the code’s guaranteed burst-correcting capability and an irreducible polynomial that characterizes the code length. However, the true burst-correcting capability of a Fire code may exceed its guaranteed burst-correcting capability, which can be thought of as the designed burst-correcting capability of the Fire code. The true burst-correcting capability of a Fire code depends on the irreducible polynomial used in code construction. In this paper, the maximum true burst-correcting capabilities of Fire codes are considered. Fire codes are classified based on three parameters: their designed burst-correcting capabilities, the least common multiple of the periods of the binomials, and the irreducible polynomials used in their constructions, and the ratios of the periods of primitive polynomials of the same degrees as the irreducible polynomials to the periods of the irreducible polynomials. It is shown that the maximum true burst-correcting capability of each class, which pertains to an infinite number of codes, can be determined by checking whether or not a finite number of incongruences have a solution. It is also shown that in each class, there is an infinite number of Fire codes, with increasing code lengths, that attain this maximum. The maximum true burst-correcting capabilities of several classes of Fire codes are determined. It is also shown that there are infinite sequences of irreducible polynomials that generate cyclic codes of rates approaching one and with burst-correcting capabilities that exceed any given number. [ABSTRACT FROM AUTHOR]

This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances. [ABSTRACT FROM PUBLISHER]

In this paper, the large family of generalized Kasami sequences has been studied. In particular, the cross-correlation distribution among these sequences has been explicitly calculated. Meanwhile, the weight distributions of two classes of cyclic codes could also be determined. This paper generalizes the results from several previous papers. [ABSTRACT FROM AUTHOR]

In this paper, we will study the exponential sum Σxϵ Fq χ(αx(Pk+1)/2 + βx) that is related to the generalized Coulter-Matthews function x(Pk+1)/2 with k/gcd (m, k) odd. As applications, we obtain the following: • the correlation distribution of a p-ary m-sequence and a decimated m-sequence of degree (Pk+1/2; • the weight distribution of the cyclic code whose dual has two zeros π-1 and xPk+1/. This paper extends the results in Ness et al. (2006). [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

Constant composition codes are codes where the frequency distribution of the elements in a codeword is the same for all codewords. In this paper, three classes of constant composition codes are constructed. These codes are subcodes of cyclic codes which have few weights occurring among the codewords. The new codes are excellent asymptotically compared to the previously best known constant composition codes. [ABSTRACT FROM AUTHOR]

In coding theory, self-dual codes and cyclic codes are important classes of codes which have been extensively studied. The main objects of study in this paper are self-dual cyclic codes over finite fields, i.e., the intersection of these two classes. We show that self-dual cyclic codes of length n over \BBFq exist if and only if n is even and q = 2^m with m a positive integer. The enumeration of such codes is also investigated. When n and q are even, there is always a trivial self-dual cyclic code with generator polynomial x^n\over 2+1. We, therefore, classify the existence of self-dual cyclic codes, for given n and q, into two cases: when only the trivial one exists and when two or more such codes exist. Given n and m, an easy criterion to determine which of these two cases occurs is given in terms of the prime factors of n, for most n. We also show that, over a fixed field, the latter case occurs more frequently as the length grows. [ABSTRACT FROM AUTHOR]

Though the entanglement-assisted formalism provides a universal connection between a classical linear code and an entanglement-assisted quantum error-correcting code (EAQECC), the issue of maintaining large amount of pure maximally entangled states in constructing EAQECCs is a practical obstacle to its use. It is also conjectured that the power of entanglement-assisted formalism to convert those good classical codes comes from massive consumption of maximally entangled states. We show that the above conjecture is wrong by providing families of EAQECCs with an entanglement consumption rate that diminishes linearly as a function of the code length. Notably, two families of EAQECCs constructed in the paper require only one copy of maximally entangled state no matter how large the code length is. These families of EAQECCs that are constructed from classical finite geometric LDPC codes perform very well according to our numerical simulations. Our work indicates that EAQECCs are not only theoretically interesting, but also physically implementable. Finally, these high performance entanglement-assisted LDPC codes with low entanglement consumption rates allow one to construct high-performance standard QECCs with very similar parameters. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the structure and properties of cyclic codes over the ring F2 + vF2. We first study the relationship between cyclic codes over F2 + vF2 and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given length over F2 + vF2. [ABSTRACT FROM AUTHOR]

In this paper, an explicit construction of binary self-dual cyclic codes of length n going to infinity with a minimal distance at least half the square root of n is presented. The same idea is also used to construct more general binary cyclic codes with a large minimal distance. Finally, in the special case of self-dual cyclic codes, a simplified version of a proof by Conway and Sloane is given, showing an upper bound for the distance of binary self-dual codes. [ABSTRACT FROM AUTHOR]

This paper shows that q-ary images of some cyclic codes can be put into concatenated form in the sense of Jensen (1985). This description utilizes an isomorphism between the GF(q)-algebras GF(q)[z]/(znm - 1) and GF(q)[x,y]/(xn- 1,ym - x). We also describe an element of the constacyclic outer code, which is of great help in determining one of its generator polynomials. [ABSTRACT FROM AUTHOR]

We give a new method in order to obtain Weil-Serre type bounds on the minimum distance of arbitrary cyclic codes over Fpe of length coprime to p, where e ≥ 1 is an arbitrary integer. In an earlier paper we obtained Weil-Serre type bounds for such codes only when e = 1 or e = 2 using lengthy explicit factorizations, which seems hopeless to generalize. The new method avoids such explicit factorizations and it produces an effective alternative. Using our method we obtain Weil—Serre type bounds in various cases. By examples we show that our bounds perform very well against Bose—Chaudhuri—Hocquenghem (BCH) bound and they yield the exact minimum distance in some cases. [ABSTRACT FROM AUTHOR]