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As a particular subclass of cyclic codes, BCH codes have wide applications in storage devices, communication systems, consumer electronics and other fields. However, parameters of BCH codes are unknown in general. In this paper, we investigate parameters of BCH codes of length q m + 1 λ where λ ∣ q + 1 .Some new techniques are employed to study the coset leaders. For any odd prime power q and m = 4 , 8 , or m ≥ 12 and m ≡ 4 (mod 8) , the second, the third and the fourth largest coset leaders modulo q m + 1 are determined, and the dimensions of some BCH codes of length q m + 1 with large designed distances are given. For 1 < λ < q + 1 , the first few largest coset leaders and the coset leaders modulo q m + 1 λ in the range 1 to q ⌊ (m + 1) / 2 ⌋ λ are studied, and the dimensions of some BCH codes of length q m + 1 λ are given as well. The BCH codes presented in this paper are LCD codes and have a sharper lower bound on the minimum distance than the well-known BCH bound. [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 investigate the algebraic structure of 2 r 8 s -additive codes, where r and s are nonnegative integers, 2 (respectively, 8 ) denotes the finite field of order 2 (respectively, 8). We first give the generator polynomials of additive cyclic codes over 8 and then the generator polynomials of additive cyclic codes over 2 8 is also given. In addition, we introduce a linear map W : 2 8 → 2 , and study its properties. What's more, the dual of additive cyclic codes over 2 8 are investigated as well. And we get that the duals of any additive cyclic codes over 2 8 are also additive cyclic codes. Finally, separable 2 8 -additive cyclic codes are investigated. [ABSTRACT FROM AUTHOR]

In [C. Ding, An infinite family of Steiner systems $ S(2, 4, 2^m) $ from cyclic codes, J. Combin. Des. 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 2 \pmod{4} \ge 6 $ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 0 \pmod{4} \ge 4 $ supported by this family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $ S(2, 4, 2^m) $ for all even $ m \geq 4 $. Furthermore, this paper also determines the parameters of other $ 2 $-designs supported by this family of extended cyclic 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]

Cyclic codes are an important subclass of linear codes, they not only have good algebraic structure, but also are easy to be encoded and decoded. At present, researchers have constructed many optimal ternary cyclic codes, but the study on quinary cyclic codes is less developed. In this paper, by analyzing the solutions of certain equations over F 5 m , we construct some optimal quinary cyclic codes with three zeros and with parameters [ 5 m - 1 , 5 m - 2 - 2 m , 4 ] , [ 5 m - 1 , 5 m - 2 - 3 m 2 , 4 ] . Moreover, the weight distributions of two classes of their duals are also provided. [ABSTRACT FROM AUTHOR]

In this paper we give three constructions of cyclic self-orthogonal codes over Z 2 k , for k ≥ 3 , from Boolean functions on n variables. The first construction for each k, 3 ≤ k ≤ n , yields a self-orthogonal Z 2 k -code of length 2 n + 2 with all Euclidean weights divisible by 2 k + 1. In the remaining two constructions, for each even n and k ≥ 3 , we generate a self-orthogonal Z 2 k -code of length 2 n + 1. All Euclidean weights in the constructed code are divisible by 2 2 k - 1 or 2 k + 1 , depending on which of the two constructions is used. [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 paper aims to investigate the behaviour of cyclic codes over a finite ring, where the ring S = Tq[v, w]/⟨v² - α², w³ - β²w, vw - wv⟩ is constructed using a field Tq of order q, with q being an odd prime power, and nonzero elements α and β from Tq. With the help of Gray images of cyclic code over the defined ring S, we construct LCD codes. A condition is also established for cyclic codes that contain their duals. Additionally, by using the self-orthogonal condition of cyclic codes over S, we are able to construct quantum codes. [ABSTRACT FROM AUTHOR]

This paper studies λ-constacyclic codes and skew λ-constacyclic codes over the finite commutative non-chain ring R = Z4 + uZ4 + u²Z4 with u3 = 0 for λ = (1 + 2u + 2u²) and (3 + 2u + 2u²). We introduce distinct Gray maps and show that the Gray images of λ-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over Z4. It is also shown that the Gray images of skew λ-constacyclic codes are quasi-cyclic codes of length 2n and index 2 over Z4. Moreover, the structure of λ-constacyclic codes of odd length n over the ring R is determined and give some suitable examples. [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]

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]

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 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]

Let $ q $ be an odd prime power and $ \mathbb{F}_q $ be the finite field with $ q $ elements. In this paper, suppose ring $ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $, where $ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $ We first give a Gray map from $ R $ onto $ \mathbb{F}_q^{4} $ and consider a decomposition of the ring $ R $. Additionally, we investigate linear complementary dual (LCD) codes over the ring $ R $. Some conditions for such linear codes over $ R $ to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over $ R $. [ABSTRACT FROM AUTHOR]

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]

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes $ \mathcal{C}_{(2,v)} $ by using monomials $ x^2 $ and $ x^v $ for some suitable $ v $ and explain the novelty of the codes. Furthermore, the weight distribution of $ \mathcal{C}_{(2,v)}^{\perp} $ for $ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $ is determined. [ABSTRACT FROM AUTHOR]

For an odd prime $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature. [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]

This paper studies cyclic and (1 + 2v)-constacyclic codes over the ring Z4[v]/hv ³ − 1i. By introducing three different Gray maps, we show that the Gray images of cyclic codes are quasi-cyclic codes over Z4 and that of (1 + 2v)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z4. Moreover, we show that the Gray image of skew (1 + 2v)-constacyclic code is a quasi-cyclic code over Z4. [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 s ≥ 1 be a fixed integer. In this paper, we focus on generating cyclic codes over the ring R (α 1 , α 2 , ... , α s) , where α i ∈ F q ∖ { 0 } , 1 ≤ i ≤ s , by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing F q reversibility. The aim to introduce the F q reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases (A , T , G , C) . Let be a fixed integer. In this paper, we focus on generating cyclic codes over the ring , where , , by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing reversibility. The aim to introduce the reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases . [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]

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]

Let $ \mathbb{Z}_4 $ be the ring of integers modulo $ 4 $. This paper studies mixed alphabets $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive cyclic and $ \lambda $-constacyclic codes for units $ \lambda = 1+2u^2,3+2u^2 $. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain $ \mathbb{Z}_4 $-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary 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]

In this paper, we classify completely the negacyclic codes of length 3 p s over p m + u p m + u 2 p m when u 3 = 0 by using six-cyclotomic codes. [ABSTRACT FROM AUTHOR]

A length ml, index l quasi-cyclic code can be viewed as a cyclic code of length m over the field Fql via a basis of the extension Fql/Fq. This cyclic code is an additive cyclic code. In [C. Güneri, F. Ozdemir, P. Sol'e, On the additive cyclic structure of quasi-cyclic codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize the (l,m) values for one-generator quasi-cyclic codes for which it is impossible to have an Fql -linear image for any choice of the polynomial basis of Fql/Fq. But this characterization for some (l,m) values is very intricate. In this paper, by the use of this characterization, we give a more simple characterization. [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]