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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 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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

It has been known for a long time that t-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a t-design. While a lot of progress in the direction of constructing codes from t-designs has been made, only a small amount of work on the construction of t-designs from codes has been done. The objective of this paper is to construct infinite families of 2-designs and 3-designs from a type of binary linear codes with five weights. The total number of 2-designs and 3-designs obtained in this paper are exponential in any odd m and the block size of the designs varies in a huge range. [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]

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]

Coding theory and t-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus–Mattson theorem and do not admit 2-transitive or 2-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed–Muller codes, we prove that they still hold 2-designs. We also determine the parameters of these 2-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and 2-designs from a special type of ternary projective codes. [ABSTRACT FROM AUTHOR]

Recently, b-symbol codes are proposed to protect against b-symbol errors in b-symbol read channels. It is an interesting subject of study to consider the complete b-symbol weight distribution of cyclic codes since b-symbol metric is a generalization for Hamming metric. The complete b-symbol Hamming weight distribution of irreducible cyclic codes is known in only a few cases. In this paper, we give a complete b-symbol Hamming weight distribution of a class of irreducible cyclic codes with two nonzero b-symbol Hamming weights. [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]

Let q = 2 m . The projective general linear group PGL (2 , q) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over GF (2 h) that are invariant under PGL (2 , q) are trivial codes: the repetition code, the whole space GF (2 h) 2 m + 1 , and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all 3 - (q + 1 , k , λ) designs that are invariant under PGL (2 , q) are determined. The second objective is to present two infinite families of cyclic codes over GF (2 m) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL (2 , q) , therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters [ q + 1 , q - 3 , 4 ] q , where q = 2 m , and m ≥ 4 is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3- (q + 1 , 4 , 2) design. A code from the second family has parameters [ q + 1 , 4 , q - 4 ] q , q = 2 m , m ≥ 4 even, and the minimum weight codewords support a 3- (q + 1 , q - 4 , (q - 4) (q - 5) (q - 6) / 60) design, whose complementary 3- (q + 1 , 5 , 1) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over GF (q) that can support a 3- (q + 1 , q - 4 , (q - 4) (q - 5) (q - 6) / 60) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound. [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]

Let p be an odd prime, and m, k and d be positive integers such that $$2 \le k\le \frac{m+1}{2}$$ and $$\hbox {gcd}(m,d)=1. \pi $$ is a primitive element of the finite field $${\mathbb {F}}_{p^{m}}$$ . The weight enumerator of cyclic codes over $${\mathbb {F}}_{p}$$ whose duals have 2 k zeros $$\pi ^{-(p^{jd}+1)/2}$$ and $$-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-1)$$ is determined in the present paper. The weight enumerator of cyclic codes over $${\mathbb {F}}_{p}$$ whose duals have $$2k-1$$ zeros $$\pi ^{-(p^{(k-1)d}+1)/2}, \pi ^{-(p^{jd}+1)/2}$$ and $$-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-2)$$ is also determined when $$2\not \mid \frac{m}{gcd(m,k-1)}$$ holds. [ABSTRACT FROM AUTHOR]

In this paper, we study the quantum codes over , which are obtained from (λ1 + λ2)-constacyclic codes over the semi local ring , where v2 = 1, p is odd prime. We decompose a (λ1 + λ2)-constacyclic code over into two constacyclic codes over such as (λ1 + λ2)-constacyclic and (λ1–λ2)-constacyclic. We give the necessary and sufficient condition that the (λ1 + vλ2)-constacyclic codes over contain their duals. We give some examples of non binary quantum codes. [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]

In this paper, a family of five-weight reducible cyclic codes is presented. Furthermore, the weight distribution of these cyclic codes is determined, which follows from the determination of value distributions of certain exponential sums. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of cyclic and skew cyclic codes over the finite ring Dk = Fq + v1Fq + ... + vk Fq, vi² = vi, vivj = vjvi = 0, 1 ≤ i,j ≤ k, q = pm, p is a prime for k ≥1 which contains the ring Fq + v1Fq, v²1 = v1. We define a new Gray map from Dk to Fkq+1. The algebraic structures of cyclic codes and duality properties are investigated. A linear code over Dk is represented by means of k + 1 q-ary codes. The non trivial automorphism over Dk is given and the skew cyclic codes over Dk are introduced. The algebraic structure of skew cyclic codes and duality properties are investigated. The Gray images of both cyclic and skew cyclic codes over Dk are obtained. [ABSTRACT FROM AUTHOR]

In this paper, a family of six-weight cyclic codes over $$\mathbb {F}_{p}$$ whose duals have three zeros is presented, where $$p$$ is an odd prime. Furthermore, the weight distributions of these cyclic codes are determined. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and m and k be two positive integers with m/gcd(m,k) being odd. This paper determines the weight distribution of a family of p-ary cyclic codes over Fp whose duals have three zeros α-2, α-(p{2k+1) and α-(p4k+1), where α is a primitive element of Fpm. [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]