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The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length n p s over p m , where p is a prime number and p does not divide n , the problem has been solved completely for all n ≤ 6 and partially for n = 7 , 8. In this paper, we solve the problem for n = 9 and all primes p different from 3 and 1 9. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length 9 p s over p m . As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes. [ABSTRACT FROM AUTHOR]

There are various metrics for researching error-correcting codes. Especially, high-density data storage system gives the existence of inconsistency for the reading and writing process. The symbol-pair metric is motivated for outputs that have overlapping pairs of symbols in a certain channel. The Rosenbloom–Tsfasman (RT) metric is introduced since there exists a problem that is related to transmission over several parallel communication channels with some channels not available for the transmission. In this paper, we determine the minimum symbol-pair weight and RT weight of repeated-root cyclic codes over R = F p m [ u ] / ⟨ u 4 ⟩ of length n = p k . For the determination, we explicitly present third torsional degree for all different types of cyclic codes over R of length n. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and m and s positive integers, with m even. Let further F p m be the finite field of p m elements and R = F p m + u F p m ( u 2 = 0 ). Then R is a finite chain ring of p 2 m elements, and there is a Gray map from R N onto F p m 2 N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over F p m by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length p s over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length p s over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. [ABSTRACT FROM AUTHOR]

This research paper discusses the construction of novel and better quantum codes from the direct product of t -copies of ring R (discussed in Section 2), using cyclic codes over R by employing the CSS construction technique. Here, we present an overview of the structure and essential properties of a ring R. Furthermore, we analyze the distance-preserving nature of the Gray map in Subsection 2.1. Here, we also investigate maximum distance separable (MDS) codes by using the concept of quantum singleton defect (QSD), which indicates the overall quality of codes. To demonstrate the practicality of our findings, we provide illustrative examples implemented using the Magma software. [ABSTRACT FROM AUTHOR]

Cryptology is a part of mathematics as encryption and decryption. The purpose of encryption is to make information incomprehensible when it is in the hands of unauthorized people. The receiver can decrypt the message that encrypted by the sender with helping of the key. The important point is that the key cannot be decrypted by other people. One Time Pad method solves this problem. The key is used only once each encryption in this method. So, the key becomes harder to guess. If the key is solved by unauthorized people, the message cannot be solved. Because of with each decryption, many meaningful messages are obtained. Every cyclic shift in a cyclic code constructs a new key and in each encryption is used the new key. Many keys are generated thanks to cyclic codes. In this paper, we improve the new encryption scheme by using the cyclic codes with One Time Pad method. [ABSTRACT FROM AUTHOR]

In this paper, we study mixed alphabet codes over two alphabets Z2, and its extension Z2[u] = Z2 + uZ2, u2 = 0. We define new Gray maps, using which we study cyclic codes and obtain the properties of their generators. Further, we construct constacyclic codes, and linear codes, and observe the useful properties of their generators. Finally, we present some examples of cyclic codes and binary linear codes. [ABSTRACT FROM AUTHOR]

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost MDS codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance 3 and a class of quantum codes with minimum distance 4 are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a theory for constructing cyclic codes of odd length over the ring R = ℤ 4 + u ℤ 4 + v ℤ 4 , where u 2 = v 2 = u v = v u = 0 , which plays an important role in DNA computing. A direct link between the elements of R and the 6 4 codons used in the amino acids of the living organisms is established. Then, we investigate reversible cyclic codes and reversible complement cyclic codes of odd length over R. Moreover, we give some properties of binary images of the codons under the Gray map. Finally, two examples of cyclic codes over R with their minimum Hamming distance will be studied as well. [ABSTRACT FROM AUTHOR]

The classical way of extending an [ n , k , d ] linear code C is to add an overall parity-check coordinate to each codeword of the linear code C. This extended code, denoted by C ‾ (− 1) and called the standardly extended code of C , is a linear code with parameters [ n + 1 , k , d ¯ ] , where d ¯ = d or d ¯ = d + 1. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. [ABSTRACT FROM AUTHOR]

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of F q P Q , where P = F q [ v ] 〈 v 3 − α 2 2 v 〉 and Q = F q [ u , v ] 〈 u 2 − α 1 2 , v 3 − α 2 2 v 〉 are nonchain finite rings and α i is in F q / { 0 } for i ∈ { 1 , 2 } , where q = p m with m ≥ 1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes. [ABSTRACT FROM AUTHOR]

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]

Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length p k over the Galois ring GR (p 2 , m) is the key to construct self-dual cyclic codes of length p k n over the integer residue class ring Z p 2 for any positive integer n satisfying gcd (p , n) = 1 . So far, existing literature has only determined the number of these self-dual cyclic codes (Des Codes Cryptogr 63:105–112, 2012). In this paper, we give an efficient construction for all distinct self-dual cyclic codes of length p k over GR (p 2 , m) by using column vectors of Kronecker products of matrices with specific types. On this basis, we further obtain an explicit expression for all these self-dual cyclic codes by using binomial coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we use a method inspired by the valuation theory to classify completely cyclic codes of length 3 p s over p m + u p m + u 2 p m . Then, we compute their separation parameters which was useful to compute the Hamming distance in [8]. [ABSTRACT FROM AUTHOR]

There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. Recently, several infinite families of 4-designs from linear codes were discovered. But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. In this paper, we construct eight 5-designs from some lifted codes of some known linear codes. • There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. • Recently, several infinite families of 4-designs from linear codes were discovered. • But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. • It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. • In this paper, we construct eight new 5-designs from some lifted codes of some known linear codes. [ABSTRACT FROM AUTHOR]

Hermitian dual-containing codes play an important role in constructing quantum codes, constructing Hermitian dual-containing codes with optimal parameters is an interesting topic. The objective of this paper is to construct several classes of quantum codes with optimal parameters. Using cyclic codes, three classes of optimal Hermitian dual-containing codes are constructed. By employing the Hermitian construction, three classes of quantum codes with optimal parameters are obtained. These codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented. [ABSTRACT FROM AUTHOR]

Let m = 8 k and α be a primitive element of the finite field G F (2 m) , where k ≥ 2 is an integer. In this paper, a class of binary cyclic codes C (u , v) of length 2 m - 1 with two nonzeros α - u and α - v is studied, where (u , v) = (1 , (2 m - 1) / 17) . It turns out that C (u , v) has parameters [ 2 m - 1 , 2 m - m - 9 , 4 ] and is distance-optimal with respect to the Sphere Packing bound. The weight distribution of the dual of C (u , v) is also completely determined based on some results on Gaussian periods. [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 paper presents a novel efficient encoding and decoding method for self‐correction and anti‐interference used in big data product tracing system based on the cyclic code theory. The principle of constructing code set is first illustrated. Then the specific encoding and decoding methods suitable for the algebra rules are invented in sequence for efficiency. At last, to test the new coding system, the paper accelerates the decoding time by OpenMP more than 36 times. The error‐correcting code presents its perfect recovery property with a recovery rate of more than 95% under 60% damage rate. In analyzing its efficiency, the encoding process only costs 0.0045 second in average while the time of decoding method will increase with the damage rate's ascending, but it can be controlled within 16 seconds. In comparison to other coding patterns in the references, the newly error‐correcting and anti‐interference coding method achieves a balance in complexity, encoding rate, and decoding time with its priority in recovering rate under the damage rate of 60%. [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]

In the quantum coding theory, quantum maximum distance separale (MDS) codes that satisfy the quantum Singleton bound have become an important research topic. However, it is not an easy task to find quantum MDS codes with minimum distance more than q + 1 . Entanglement-assisted quantum MDS (EAQMDS) codes form an important family of quantum codes that can improve the minimum distance of quantum MDS codes to exceed q + 1 . Let q be an odd prime power. In this paper, using a formula on the number of maximally entangled states, we construct three classes of EAQMDS codes of length n = q 2 - 1 λ 1 λ 2 , where λ 1 and λ 2 are odd factors of q - 1 and q + 1 , respectively. Most of these EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature. Moreover, the minimum distance of these resulting codes is larger than q + 1 . [ABSTRACT FROM AUTHOR]

Let m ≥ 1 be a fixed integer and q be an odd prime such that q = p m . The aim of this paper is to study cyclic codes over a non-chain ring R = F q + v F q + v 2 F q , where v 3 = v . Precisely, we describe better quantum error-correcting codes than the previously known quantum error-correcting codes over F q . Moreover, as an application, we construct MDS LCD codes and prove that the Gray image of an LCD code of length n over R is also an LCD code of length 3n over F 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]

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, by investigating the solutions of certain equations over finite fields, we make progress towards three conjectures about optimal ternary cyclic codes which were proposed by Ding and Helleseth. [ABSTRACT FROM AUTHOR]

Encryption is the process of scrambling a message and can provide a means of securing information. Information security is becoming more important in data storage and transmission. It is known that most encryption method in the literature. In this paper we propose two new encryption schemes by using cyclic codes. Our method is based on the One Time Pad system. We use the properties of cyclic codes to provide its security. [ABSTRACT FROM AUTHOR]

In this paper, we investigate all coset leaders of primitive BCH codes for δ in the range 1 ≤ δ ≤ q m + 7 2 , which extends Liu and Shi's results. Besides, we also generalize Shi's results by proposing the maximum designed distance of non-narrow-sense( b = k 2 q 2 + k 1 q + k 0 ) primitive BCH codes which can contain their Euclidean dual. At the end, we calculate the dimension of the Euclidean dual containing non-narrow-sense primitive BCH codes and construct some quantum BCH codes. [ABSTRACT FROM AUTHOR]

Interplay between coding theory and combinatorial t-designs has attracted a lot of attention. It is well known that the supports of all codewords of a fixed Hamming weight in a linear code may hold a t-design. In this paper, we first settle the weight distributions of two classes of linear codes, and then determine the parameters of infinite families of 2-designs held in these codes. [ABSTRACT FROM AUTHOR]

The Fletcher checksum (FC) is a means of error detection that is analogous to cyclic redundancy checking but easier to implement in software. Now FC actually has two versions, one is Fletcher’s original paper version termed as FC and the other is RFC1146 version termed as FC-RFC. This paper shows that the FC is a cyclic code. This is done by introducing a cyclic code known as the cyclic Fletcher code (CFC) and then showing that the CFC is equivalent to the FC. The clear algebraic structure of cyclic codes is used to analyze the code block length and code distance, and then the single error correction, double error detection, and burst error detection of the CFC are analyzed. Also given is the mathematical relationship between the CFC and a modified FC. Finally, the issue of implementation is considered, and a new scheduling for the CFC is proposed to prevent data hazards. A simulation shows that the proposed scheduling offers improvements of at least 46% in throughput for the CFC and FC-RFC. [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, two classes of cyclic codes whose duals have two zeros are presented. The weight distributions of these cyclic codes are settled with the help of Gaussian periods. The duals of one class of cyclic codes are also studied. Some of the cyclic codes presented in this paper are optimal in the sense that they meet some bounds of linear codes. [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]

For any odd positive integer n, we express cyclic codes over Z 4 of length 4n in a new way. Based on the expression of each cyclic code C , we provide an efficient encoder and determine the type of C . In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic codes over Z 4 of length 4n and correct a mistake in the paper "Concatenated structure of cyclic codes over Z 4 of length 4n" (Cao et al. in Appl Algebra Eng Commun Comput 10:279–302, 2016). In addition, we obtain 50 new self-dual cyclic codes over Z 4 of length 28. [ABSTRACT FROM AUTHOR]

In this paper, we first study a general type of affine‐ invariant codes and their support 2‐designs. Then we derive infinite families of 2‐designs from a special class of affine‐invariant codes and obtain their parameters by determining the weight distribution of the linear codes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a construction of q-ary quantum codes from Hermitian self-orthogonal cyclic codes over the finite chain ring R = F q 2 [ u ] / ⟨ u k ⟩ , with u k = 0 , where F q 2 is a finite field with q 2 elements, q = p m , p a prime. A Gray map from R to F q 2 k is defined. Some characterizations of cyclic codes over R have been given in terms of their different types of generators. The structure of their dual codes has also been determined, and a necessary and sufficient condition for these codes to be self-orthogonal is presented. The construction of quantum codes is derived by applying Hermitian construction to the Gray images of self-orthogonal cyclic codes over R. From this construction, we have been able to obtain some new quantum codes with better parameters than some presently known comparable best codes available in the literature. Some of these codes have been given as examples, and the others have been given in the form of two tables. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic codes over a finite non-chain ring F p m + v F p m with v2 = 1 and find new and better quantum error-correcting codes than previously known quantum error correcting codes over F p m . Therewith, we characterize the LCD codes and obtain many new LCD codes. Also, we prove that the Gray map of an LCD code of length n over F p m + v F p m is an LCD code of length 2n over F p m . [ABSTRACT FROM AUTHOR]

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF ( q ) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] for all m ≥ 2 . Their extended codes have parameters [ 3 m , 3 m − 1 − 2 m , 5 ] for all m ≥ 2 , and are also optimal. [ABSTRACT FROM AUTHOR]

With entanglement-assisted (EA) formalism, arbitrary classical linear codes are allowed to transform into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, based on classical cyclic MDS codes by exploiting pre-shared maximally entangled states, we construct two families of q-ary entanglement-assisted quantum MDS codes q 2 + 1 a q 2 + 1 a − 2 d − 1 + c d c , where q is a prime power in the form of am + l, and a = (l2 + 1) or a = l 2 + 1 5 . We show that all of q-ary EAQMDS have minimum distance upper limit much larger than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

Let p and q be odd primes and q be a cubic primitive modulo p u for some positive integer u. In this paper, we prove that the solutions of some Diophantine equations provide the weight enumerators of some cubic primitive irreducible cyclic codes of prime length. Bounds on the minimum distances of these codes are also given. [ABSTRACT FROM AUTHOR]

For a simple-root λ -constacyclic code C over F q , let 〈 ρ 〉 and 〈 ρ , M 〉 be the subgroups of the automorphism group of C generated by the cyclic shift ρ , and by the cyclic shift ρ and the scalar multiplication group M , respectively. Let N G (C ⁎) be the number of orbits of a subgroup G of the automorphism group of C acting on C ⁎ = C ﹨ { 0 }. In this paper, we establish explicit formulas for N 〈 ρ 〉 (C ⁎) and N 〈 ρ , M 〉 (C ⁎). Consequently, we derive a upper bound on the number of non-zero weights of C. We present some irreducible and reducible λ -constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee N 〈 ρ 〉 (C ⁎) = N 〈 ρ , M 〉 (C ⁎) is presented. [ABSTRACT FROM AUTHOR]

As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal p-ary cyclic codes with parameters [pm-1,pm-3m2-2,4]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[p^m-1, p^m-\frac{3m}{2}-2, 4]$$\end{document} by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code C(0,1,e)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {C}_{(0,1,e)}$$\end{document} and present three classes of optimal codes with parameters [7m-1,7m-2m-2,4]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[7^m-1,7^m-2m-2,4]$$\end{document}. Additionally, we provide the weight distribution of one class of their duals. [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. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [26] and the works in Liu et al. (2023) [15] and Liu et al. (2023) [16] , the objectives of this paper are the construction and analysis of four infinite families of ternary cyclic codes with length n = 3 m − 1 for odd m and dimension k ∈ { n / 2 , (n + 2) / 2 } whose minimum distances have a square-root-like lower bound. Their duals have parameters [ n , k ⊥ , d ⊥ ] , where k ⊥ ∈ { n / 2 , (n − 2) / 2 } and d ⊥ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes. [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]

Entanglement-assisted quantum error-correcting (EAQEC) codes could generalize and improve performance of standard quantum error-correcting (QEC) codes to a great extent. In this paper, series of EAQEC codes of length n = q - 1 a (q + 1) are constructed from cyclic codes and negacyclic codes, where q is a prime power and a is a positive integer such that a ∣ (q - 1) . It turns out that the number of required entanglement bits can take almost all possible values. Consequently, our EAQEC codes have flexible parameters and most of them are new. For given the same length, our construction contain and extend those known consequences in Grassl et al. (Int J Quantum Inf 2(1):55–64, 2004), Jin et al. (IEEE Trans Inf Theory 56:4735–4740, 2010), Kai et al. (IEEE Trans Inf Theory 60:2080–2086, 2014), Jin and Xing (IEEE Trans Inf Theory 60:2921–2925, 2014), Chen et al. (IEEE Trans Inf Theory 61:1474–1484, 2015), Zhang and Ge (IEEE Trans Inf Theory 61:5224–5228, 2015; Des Codes Cryptogr 83(3):503–517, 2016), Shi et al. (Cryptogr Commun 10(6):1165–1182, 2018; Finite Fields Appl 46:347–362, 2017), Fan et al. (Quantum Inf Comput 16:423–434, 2016), Lu et al. (Quantum Inf Process 17(69):1–23, 2018), Li et al. (Int J Quantum Inf 17(1):1950022, 2019), Liu et al. (Quantum Inf Process 17(210):1–19, 2018), Fang et al. (Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs. arXiv:https://arxiv.org/abs/1812.09019v3). Above all, all our codes are maximum-distance-separable (MDS) if their minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]

Combinatorial t‐designs have wide applications in coding theory, cryptography, communications, and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a t‐design. In this paper, we first determine the weight distributions of a class of linear codes derived from the dual of some extended cyclic codes. We then obtain infinite families of 2‐designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes. By a simple counting argument, we obtain exponentially many 2‐designs. [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]