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Let F 2 m be a finite field of cardinality 2 m and s a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over F 2 m , an efficient method for the construction of all distinct self-dual cyclic codes with length 2 s over the finite chain ring F 2 m + u F 2 m ( u 2 = 0 ) is provided. On that basis, an explicit representation for every self-dual cyclic code of length 2 s over F 2 m + u F 2 m and an exact formula to count the number of all these self-dual cyclic codes are given.

Let F q be a finite field and n a positive integer. In this paper, we find a new combinatorial method to determine weight enumerators of reducible cyclic codes and their dual codes of length n over F q , which just generalize results of Zhu et al. (2015); especially, we also give the weight enumerator of a cyclic code, which is viewed as a partial Melas code. Furthermore, weight enumerators obtained in this paper are all in the form of power of a polynomial.

Let D 2 n = 〈 x , y ∣ x n = 1 , y 2 = 1 , y x y = x − 1 〉 be a dihedral group, and R = GR ( p 2 , m ) be a Galois ring of characteristic p 2 and cardinality p 2 m where p is a prime. Left ideals of the group ring R [ D 2 n ] are called left dihedral codes over R of length 2 n , and abbreviated as left D 2 n -codes over R . Let gcd ( n , p ) = 1 in this paper. Then any left D 2 n -code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A i and outer codes C i , where A i is a cyclic code over R of length n and C i is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R . Specifically, a generator matrix and basic parameters for each outer code C i are given. A formula to count the number of these codes is obtained and the dual code for each left D 2 n -code is determined. Moreover, all self-dual left D 2 n -codes and self-orthogonal left D 2 n -codes over R are presented.

In this paper, R is a finite chain ring with residue field F q and γ is a unit in R . By assuming that the multiplicative order u of γ is coprime to q , we give the trace-representation of any simple-root γ -constacyclic code over R of length l , and on the other hand show that any cyclic code over R of length u l is a direct sum of trace-representable cyclic codes. Finally, we characterize the simple-root, contractable and cyclic codes over R of length u l into γ -constacyclic codes of length l .

Let m and k be two positive integers such that v 2 ( m ) = v 2 ( k ) , where v 2 ( ⋅ ) denotes the 2-valuation function, and let α be a primitive element of F p 2 m . Let C be a cyclic code over F p with a parity-check polynomial h 1 ( x ) h 2 ( x ) h 3 ( x ) ∈ F p 2 m [ x ] , where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of α − p k + 1 2 , − α − p k + 1 2 and α − p m + 1 2 over F p respectively. In this paper, we determine the weight distribution and the complete weight distribution of the code C .

The aim of this paper is to establish all self-dual -constacyclic codes of length ps over the finite commutative chain ring R=Fpm+uFpm, where p is a prime and u2=0. If =+u for nonzero elements , of Fpm, the ideal u is the unique self-dual (+u)-constacyclic codes. If = for some nonzero element of Fpm, we consider two cases of . When =1, i.e., =1 or 1, we first obtain the dual of every cyclic code, a formula for the number of those cyclic codes and identify all self-dual cyclic codes. Then we use the ring isomorphism to carry over the results about cyclic accordingly to negacyclic codes. When 1, it is shown that u is the unique self-dual -constacyclic code. Among other results, the number of each type of self-dual constacyclic code is obtained.

Let F q be a finite field of order q, a prime power integer, such that q = e t + 1 where t ≥ 1 , e ≥ 2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring R e , q = F q [ u ] / 〈 u e − 1 〉 . We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal F q -linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when gcd ⁡ ( n , q ) ≠ 1 and gcd ⁡ ( n , q ) = 1 , respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image φ ( C ) is an LCD code of length en over F q . Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over R e , q .

The interplay between coding theory and t -designs started many years ago. While every t -design yields a linear code over every finite field, the largest t for which an infinite family of t -designs is derived directly from a linear or nonlinear code is t = 3 . Sporadic 4 -designs and 5 -designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of 2 -designs and 3 -designs from linear codes. The parameters of some known t -designs are also derived. In addition, many conjectured infinite families of 2 -designs are also presented.

BCH codes have been studied for over fifty years and widely employed in consumer devices, communication systems, and data storage systems. However, the dimension of BCH codes is settled only for a very small number of cases. In this paper, we study the dimensions of BCH codes over finite fields with three types of lengths n , namely n = q m − 1 , n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 . For narrow-sense primitive BCH codes with designed distance δ , we investigate their dimensions for δ in the range 1 ≤ δ ≤ q ⌈ m 2 ⌉ + 1 . For non-narrow sense primitive BCH codes, we provide two general formulas on their dimensions and give the dimensions explicitly in some cases. Furthermore, we settle the minimum distances of some primitive BCH codes. We also explore the dimensions of the BCH codes of lengths n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 over finite fields.

Bierbrauer (2012) developed the theory of q-linear cyclic codes over (Fq)m and he obtained a parametric description of such codes by cyclotomic cosets. Recently, Cao et al. (2015) obtained the structure of cyclic additive codes over the Galois ring GR(p,m), where m is a prime integer. Let R be a finite commutative ring and Rn=R[x]xn1. In this paper, we generalize the theory of Fq-linear codes over vector spaces to R-linear codes over free R-algebras (free as R-module). We call these codes, R-additive codes. We introduce a one-to-one correspondence between the classes of cyclic R-additive code and the classes of Rn-linear code. Using the structure of Rn-linear codes, we present the structure of cyclic R-additive codes, where R is a chain ring. Among other results, q-linear cyclic codes over (Fq)m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR(p,m) is given for an arbitrary integer m, not necessarily a prime number.

Necessary and Sufficient conditions for an equidistant code to be a 2-TA code are obtained. An explicit construction method is proposed to obtain linear MDS p + 1 , 2 , p codes over the finite field F p , where p is a prime. These codes can be used as 2-TA codes for p 2 . In particular, for p = 3 , it is observed that the linear 4, 2, 3 MDS code contradicts a result of Jin and Blaum (2007). The correct version of this result and its proof is given. Existence of some infinite families of equidistant 2-TA codes is shown by using Jacobsthal and Hadamard matrices. Some of these codes are also observed to be good equidistant code (Sinha et al., 2008).

Let F 2 m be the finite field of 2 m elements and s be any positive integer. The existing literature only gives an effective calculation method to represent all distinct Euclidean self-dual cyclic codes of length 2 s over the finite chain ring F 2 m + u F 2 m ( u 2 = 0 ) , such as in Cao et al., (2019). As a development of this topic, we provide an explicit expression for each of these self-dual cyclic codes, using binomial coefficients. The Gray image of any self-dual cyclic code over F 2 m + u F 2 m of length 2 s is a self-dual 2 -quasi-cyclic code over F 2 m of length 2 s + 1 . In particular, we give a generator matrix for each of these self-dual 2 -quasi-cyclic codes over F 2 m .

Let p be an odd prime, and k an integer such that k ∣ ( p − 1 ) . This paper studies quantum codes from skew cyclic codes over the ring R = F q [ u ] 〈 u k + 1 − u 〉 , where q is a power of the prime p . We construct a set of orthogonal idempotents of the ring R and using the set, skew cyclic codes over the ring R are decomposed as direct sum of skew cyclic codes over F q . We obtain a necessary and sufficient condition for a skew cyclic code to contain its dual. As an application, we construct quantum codes from skew cyclic codes over F q . It is observed that some quantum codes we constructed are MDS quantum codes.

Finding the weight distributions of specific cyclic codes has been an interesting area of research. Although the class of cyclic codes has been studied for many years, their weight distributions are known only for a few cases. In this paper, we use matrix method to determine the weight distribution of a family of cyclic codes.

In this paper we find the complete weight hierarchy of the codes parameterized by the edges of the cycle C 4 and an upper bound in the case of the complete bipartite graph K 2 , n . Moreover we find some bounds for the generalized Hamming weights of some codes parameterized by a set of monomials of the same degree. These bounds work in the case of codes parameterized by the edges of any simple graph G . Also we establish the relationships among the generalized Hamming weights of any parameterized code of the form C X ( d ) and the generalized Hamming weights of the parameterized code of the form C X ' ( d ) , when X ' is embedded in X .

Let p be any odd prime number, m and s be arbitrary positive integers, and let F p m be the finite field of cardinality p m . Existing literature only determines the number of all (Euclidean) self-dual cyclic codes of length p s over finite chain ring R = F p m + u F p m ( u 2 = 0 ) , such as Dinh et al. (2018). Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over F p with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length p s over R . Moreover, we provide an efficient method to construct every self-dual cyclic code of length p s over R precisely.

We study α -constacyclic codes over the Frobenius non-chain ring R ≔ Z 4 [ u ] ∕ 〈 u 2 − 1 〉 for any unit α of R . We obtain new MDR cyclic codes over Z 4 using a close connection between α -constacyclic codes over R and cyclic codes over Z 4 . We first explicitly determine generators of all α -constacyclic codes over R of odd length n for any unit α of R . We then explicitly obtain generators of cyclic codes over Z 4 of length 2 n by using a Gray map associated with the unit α . This leads to a construction of infinite families of MDR cyclic codes over Z 4 , where a MDR code means a maximum distance with respect to rank code in terms of the Hamming weight or the Lee weight. We obtain 202 new cyclic codes over Z 4 of lengths 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 and 54 by implementing our results in Magma software; some of them are also MDR codes with respect to the Hamming weight or the Lee weight.

Let R = GR ( p ? , l ) be a Galois ring of characteristic p ? and cardinality p ? l , where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R . This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Then we give the trace dual code for each additive cyclic code over R from its canonical form decomposition and linear codes of length l over some extension Galois rings of Z p ? .

Let F q be a finite field with q elements, l an odd prime, and t , v positive integers such that l v | ( q - 1 ) and gcd ( t , l ) = 1 . In this paper, we give a combinatorial result and use it to determine the weight distribution of a cyclic code of length t l m with t | ( q - 1 ) , which is an open question in Yang et?al. (2013). Moreover we compute the weight distributions of cyclic codes of length t l m , where q ? 3 ( mod 4 ) and t = 4 or 8 .

In this paper, the explicit expressions for the generating idempotents, check polynomials and the parameters of all minimal cyclic codes of length t p n over F q are obtained, where p is an odd prime different from the characteristic of F q , t and n are positive integers with t ∣ ( q − 1 ) , gcd ( t , p ) = 1 and ord t p n ( q ) = ϕ ( p n ) . Our results generalize the main results in Pruthi and Arora (1997) and Arora and Pruthi (1999), which considered the cases t = 1 and t = 2 respectively. We propose an approach different from those in Pruthi and Arora (1997) and Arora and Pruthi (1999) to obtain the generating idempotents.

Irreducible cyclic codes are an interesting type of codes and have applications in space communications. They have been studied for decades and a lot of progress has been made. The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes.

We introduce a Gray map from F"p+vF"p to F"p^2 and study (1-2v)-constacyclic codes over F"p+vF"p, where v^2=v. It is proved that the image of a (1-2v)-constacyclic code of length n over F"p+vF"p under the Gray map is a distance-invariant linear cyclic code of length 2n over F"p. The generators of such constacyclic codes for an arbitrary length are determined and their dual codes are also discussed.

It is shown that there exists a perfect one-error-correcting binary code with a kernel which is not contained in any Hamming code.

Caps in a finite projective geometry over GF(4) are used for the construction of some quantum error-correcting codes, including an optimal @?27,13,[email protected]? code.

Brualdi et al. introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Hamming code to be a double-error-correcting perfect P-code. Our study is motivated by this example. In this paper we classify all poset structures which admit the extended binary Hamming code to be a double or triple-error-correcting perfect P-code.

In this note, we give the classification of self-dual F7-codes of length 12 and maximal self-orthogonal codes of lengths 10, 11 and 13. It is also shown that there is no self-dual [16,8,d⩾8] code over F7.

A code is called t-identifying if the sets B t ( x )∩C are all nonempty and different. Constructions of 1-identifying codes and lower bounds on the minimum cardinality of a 1-identifying code of length n are given. For example, we construct a 1-identifying code of length 7 with 32 codewords and show that it is optimal.

The coset graph of a nondegenerate cyclic code is orbital regular. This yields a lower bound on its average distance, a parameter which measures the average distortion of a code used in data compression. Using results of Shahrokhi and Szekely we generalize this bound to binary codes with a transitive automorphism group.

It is well known that cyclic linear codes of length n over a (finite) field F can be characterized in terms of the factors of the polynomial x n − 1 in F [ x ]. This paper investigates cyclic linear codes over arbitrary (not necessarily commutative) finite rings and proves the above characterization to be true for a large class of such codes over these rings.

We here introduce a new formalism for describing concatenated codes. Using this formalism, we show how any generalized concatenated code can be viewed as a first order concatenated code. Finally, we give an illustrative example: using Jensen's result (1985) which shows that any abelian code has a generalized concatenated structure, we first give the representation of the [49, 18, 12] abelian code introduced by Camion (1971) as a second order concatenated code; then using our description, we show that this code is also equal to the first order concatenation of two linear cyclic codes.

We give a general context in which a binary code C admits at least one code whose weight enumerator is dual to that of C. We study two examples of applications of this general result. The first one is the formal duality between the Kerdock and the generalized Preparata codes of same length and the second one gives a new example of dual weight enumerators.

Completely regular codes are a large class of codes which share many of the most interesting properties of perfect codes. This paper shows that the basic theory—including Lloyd's theorem—can be obtained very elegantly by methods inspired from the theory of distance regular graphs.

The paper contains a proof that all binary linear [18, 9, 6] codes are equivalent to the extended quadratic residue code of length 18. In addition, a complete description of the words and cosets of this code is given in terms of the special projective structure of the projective line over F 17 .

We study a construction method introduced by Kamps and van Lint and generalized by Blokhuis and Lam, and van Lint jr and show how the theorem of Cauchy and Davenport and other related results about finite fields can be used in the lengthening of covering codes over prime power alphabets.

In traditional discussions of Fire codes, the proof of the burst-correction capability of this code is rather technical. It is never left out, because the descriptions of the standard decoding algorithm (by means of shift registers) do not give an independent proof of this burst-correction capability. Here, an algebraic method of decoding Fire codes is presented, that at the same time yields a transparent proof of the burst-correction capability of Fire codes.