Search

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]

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, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth [5]. Let C (1 , e) be a cyclic code of length 3 m − 1 over GF (3) with two nonzeros α and α e , where m and e are given integers, and α is a generator of GF (3 m) ⁎. It is shown that C (1 , e) is optimal with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] if one of the following conditions is met. 1) m ≡ 0 (mod 4) , m ≥ 4 , and e = 3 m 2 + 5. 2) m ≡ 2 (mod 4) , m ≥ 6 , and e = 3 m + 2 2 + 5. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of linear codes over finite fields, viz. skew multi-twisted codes (or skew generalized quasi-twisted codes), which is a generalization of some well-known classes of linear codes such as cyclic codes and quasi-cyclic codes. We also study algebraic structures of skew multi-twisted codes and their Galois duals (i.e., orthogonal complements with respect to the Galois inner product). We also view skew multi-twisted codes as direct sums of certain concatenated codes, which gives rise to a method to construct these codes. We also determine a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Apart from this, we determine the parity-check polynomial of each skew multi-twisted code, and obtain BCH type bounds on their minimum Hamming distances. We also determine generating sets of Galois duals of some skew multi-twisted codes from generating sets of these codes. We also derive necessary and sufficient conditions under which a skew multi-twisted code is (i) Galois self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD (linear with complementary dual). As an application of this result, we also provide enumeration formulae for all Hermitian self-dual, Hermitian self-orthogonal, Hermitian LCD and Euclidean LCD multi-twisted codes over finite fields. We also obtain many linear codes with best known and optimal parameters from 1-generator skew multi-twisted codes over finite fields F 8 and F 9. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative chain ring with unique maximal ideal 〈 γ 〉 , and let n be a positive integer relatively prime to the characteristic of R / 〈 γ 〉 . In this paper, some new necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes of length n over R are provided. Explicit enumeration formulas for self-dual cyclic codes over R are studied. In particular, several main results of [21] are special cases of the present paper. [ABSTRACT FROM AUTHOR]

In this paper, we give an improved bound for mixed exponential sums associated to good cyclic codes. [ABSTRACT FROM AUTHOR]

Abstract Let m 0 and m 1 be two odd positive integers. In this paper, the weight distribution of Z 2 -double cyclic codes of length m 0 + m 1 for a special case are determined explicitly. [ABSTRACT FROM AUTHOR]

Abstract Subspace codes and particularly cyclic subspace codes have attracted a lot of research due to their applications in error correction for random network coding. In this paper, we explore the parameters of a class of the cyclic subspace codes whose subspace polynomials are more generalized. Therefore, some corollaries that extend the previously known results are obtained. Finally, a construction of k -dimensional cyclic subspace codes of size q N − 1 q − 1 and minimal distance 2 k − 2 is provided. [ABSTRACT FROM AUTHOR]

Abstract Under research for nearly sixty years, Bose–Chaudhuri–Hocquenghem (BCH) codes have played increasingly important roles in many applications such as communication, data storage and information security. However, the dimension and minimum distance of BCH codes have been seldom solved by now because of their intractable characteristics. The objective of this paper is to study the dimensions of some binary BCH codes with length n = 2 m + 1. Many new techniques are employed to investigate the coset leaders modulo n. For m = 2 t + 1 , 4 t + 2 , 8 t + 4 and m ≥ 10 , the first five largest coset leaders modulo n are determined, and the dimensions of some BCH codes of length n with designed distance δ > 2 ⌈ m 2 ⌉ are presented. These new skills and results may be helpful to study other types of cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]

Abstract Self-reciprocal and self-conjugate-reciprocal polynomials over finite fields have been of interest due to their rich algebraic structures and wide applications. Self-reciprocal irreducible monic factors of x n − 1 over finite fields and their applications have been quite well studied. In this paper, self-conjugate-reciprocal irreducible monic (SCRIM) factors of x n − 1 over finite fields of square order are focused on. The characterization of such factors is given together with the enumeration formula. In many cases, recursive formulas for the number of SCRIM factors of x n − 1 are given as well. As applications, Hermitian complementary dual cyclic codes over finite fields and Hermitian self-dual cyclic codes over finite chain rings of prime characteristic are discussed. [ABSTRACT FROM AUTHOR]

Abstract Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. The objective of this paper is to present a family of optimal ternary cyclic codes from the Niho-type exponent. Specifically, for an odd integer m and a positive integer r with 4 r ≡ 1 (mod m) , a family of cyclic codes C (u , v) of length 3 m − 1 over GF (3) with two nonzeros β u and β v is studied, where β is a generator of GF (3 m) ⁎ , u = (3 m + 1) / 2 and v = 3 r + 2 is the ternary Niho-type exponent. The parameters of this family of cyclic codes are determined. It turns out that C (u , v) is optimal with respect to the Sphere Packing bound if 9 ∤ m and otherwise almost optimal. Thanks to a recent proof of the Dobbertin–Helleseth–Kumar–Martin conjecture by Katz and Langevin, the dual of this family of cyclic codes is shown to have at most five nonzero weights. [ABSTRACT FROM AUTHOR]

Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Gröbner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. The main purpose of this paper is to establish and exploit the algebraic structure of some repeated-root two dimensional constacyclic codes of length 2 p s . 2 k over the finite field F p m . Moreover, we study all self-dual repeated-root two-dimensional ( λ , 1 ) -constacyclic codes of length 2 p s . 2 or 2 p s . 2 2 over F p m for λ ∈ { − 1 , 1 } . [ABSTRACT FROM AUTHOR]

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann–Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports. [ABSTRACT FROM AUTHOR]

Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k -Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k -Galois LCD codes is obtained, two classes of k -Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ -constacyclic codes being k -Galois LCD codes are characterized. Some classes of k -Galois LCD λ -constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ -constacyclic codes, and present a class of Hermitian LCD λ -constacyclic MDS codes. [ABSTRACT FROM AUTHOR]

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z 2 Z 4 -additive codes, Z 2 Z 2 [ u ] -linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over Z 2 Z 2 [ u 3 ] where Z 2 [ u 3 ] = { 0 , 1 , u , 1 + u , u 2 , 1 + u 2 , u + u 2 , 1 + u + u 2 } is an 8-element ring with u 3 = 0 . We study and determine the algebraic structures of linear and cyclic codes defined over this family. First, we introduce Z 2 Z 2 [ u 3 ] -linear codes and give standard forms of generator and parity-check matrices and later we present generators of both cyclic codes and their duals over Z 2 Z 2 [ u 3 ] . Further, we present some examples of optimal binary codes which are obtained through Gray images of Z 2 Z 2 [ u 3 ] -cyclic codes. [ABSTRACT FROM AUTHOR]

Cyclic Reed–Solomon codes, a type of BCH codes, are widely used in consumer electronics, communication systems, and data storage devices. This fact demonstrates the importance of BCH codes – a family of cyclic codes – in practice. In theory, BCH codes are among the best cyclic codes in terms of their error-correcting capability. A subclass of BCH codes are the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The objective of this paper is to determine the dimension and minimum distances of two classes of narrow-sense primitive BCH codes with designed distances δ = ( q − 1 ) q m − 1 − 1 − q ⌊ ( m − 1 ) / 2 ⌋ and δ = ( q − 1 ) q m − 1 − 1 − q ⌊ ( m + 1 ) / 2 ⌋ . The weight distributions of some of these BCH codes are also reported. As will be seen, the two classes of BCH codes are sometimes optimal and sometimes among the best linear codes known. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. GHWs are of great interest in many applications since they convey detailed information of linear codes. In this paper, we continue the work of [10] to study the GHWs of a family of cyclic codes with arbitrary number of nonzeroes. The weight hierarchy is determined by employing a number-theoretic approach. [ABSTRACT FROM AUTHOR]

Let α be a primitive element of a finite field F r , where r = q m 1 m 2 and gcd ⁡ ( m 1 , m 2 ) = d , so α 1 = α r − 1 q m 1 − 1 and α 2 = α r − 1 q m 2 − 1 are primitive elements of F q m 1 and F q m 2 , respectively. Let e be a positive integer such that gcd ⁡ ( e , q m 2 − 1 q d − 1 ) = 1 , F q m 2 = F q ( α 2 e ) , and α 1 and α 2 e are not conjugates over F q . We define a cyclic code C = { c ( a , b ) : a ∈ F q m 1 , b ∈ F q m 2 } , c ( a , b ) = ( T 1 ( a α 1 i ) + T 2 ( b α 2 e i ) ) i = 0 n − 1 , where T i denotes the trace function from F q m i to F q for i = 1 , 2 . In this paper, we use Gauss sums to investigate the weight distribution of C , which generalizes the results of C. Li and Q. Yue in [13,14] . Furthermore, we explicitly determine the weight distribution of C if d = 1 , 2 . Moreover, we prove it is optimal three-weight achieving the Griesmer bound if d = 1 and gcd ⁡ ( m 2 − e m 1 , q − 1 ) = 1 . [ABSTRACT FROM AUTHOR]

Cyclic codes have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms. In this paper, for an odd prime p , we investigate two classes of p -ary cyclic codes for special cases and determine their weight distributions explicitly. The results show that both codes have at most five nonzero weights. [ABSTRACT FROM AUTHOR]

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with p 4 elements, ( p a prime) are given. Second, the number and structure of γ -constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined. [ABSTRACT FROM AUTHOR]

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we give an sufficient and necessary condition that a polynomial is irreducible over F q . For different odd primes k , l and p , we obtain generator polynomials of constacyclic codes of length k l a p b over finite field F q , where char F q = p and ( ord k q , l ) = 1 . And we answer the question in [3] under the condition. [ABSTRACT FROM AUTHOR]

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes. [ABSTRACT FROM AUTHOR]

In this paper, by analyzing the solutions of certain equations over F 3 m , we present four classes of optimal ternary cyclic codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] . It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results. [ABSTRACT FROM AUTHOR]

Constacyclic codes form a well-known class of linear codes, and are generalizations of cyclic and negacyclic codes. In this paper, we determine generator polynomials of all constacyclic codes of length 4 ℓ m p n over the finite field F q with q elements, where p , ℓ are distinct odd primes, q is a power of p and m , n are positive integers. We also determine their dual codes, and list all self-dual constacyclic codes of length 4 ℓ m p n over F q . [ABSTRACT FROM AUTHOR]

Quasi-twisted codes are generalizations of the familiar linear quasi-cyclic codes. In this paper, an algebraic method is used to investigate the relationship between quasi-twisted codes and constacyclic codes. Moreover, a generator polynomial of a quasi-twisted code with constacyclic constituent codes is given. Meanwhile, the conditions for a quasi-twisted code C of index ℓ and length ℓm to be equivalent to a constacyclic code of length ℓm are obtained. Finally, some examples are presented to illustrate the discussed results. [ABSTRACT FROM AUTHOR]

Let R = Z 4 be the integer ring mod 4. A double cyclic code of length ( r , s ) over R is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . In this paper, we determine the generator polynomials of this family of codes as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Further, we also give the minimal generating sets of this family of codes as R -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic codes and constacyclic codes with shift constant ( 2 + u ) over R = Z 4 + u Z 4 , where u 2 = 1 . We determine the form of the generators of the cyclic codes over this ring and their spanning sets. Considering their Z 4 images, we prove that the Z 4 -image of a ( 2 + u ) -constacyclic code of odd length is a cyclic code over Z 4 . We also present many examples of cyclic codes over R whose Z 4 -images have better parameters than previously best-known Z 4 -linear codes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to determine the algebraic structures of all λ -constacyclic codes of length 2 p s over the finite commutative chain ring F p m + u F p m , where p is an odd prime and u 2 = 0 . For this purpose, the situation of λ is mainly divided into two cases separately. If the unit λ is not a square and λ = α + u β for nonzero elements α , β of F p m , it is shown that the ambient ring ( F p m + u F p m ) [ x ] / 〈 x 2 p s − ( α + u β ) 〉 is a chain ring with the unique maximal ideal 〈 x 2 − α 0 〉 , and thus ( α + u β ) -constacyclic codes are 〈 ( x 2 − α 0 ) i 〉 for 0 ≤ i ≤ 2 p s . If the unit λ is not a square and λ = γ for some nonzero element γ of F p m , such λ -constacyclic codes are classified into 4 distinct types of ideals. The detailed structures of ideals in each type are provided. Among other results, the number of codewords and the dual of every λ -constacyclic code are obtained. [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. Let m = 2 ℓ + 1 for an integer ℓ ≥ 1 and π be a generator of GF ( 3 m ) ⁎ . In this paper, a class of cyclic codes C ( u , v ) over GF ( 3 ) with two nonzeros π u and π v is studied, where u = ( 3 m + 1 ) / 2 , and v = 2 ⋅ 3 ℓ + 1 is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over GF ( 3 m ) , the cyclic code C ( u , v ) is shown to have minimal distance four, which is the best minimal distance for any linear code over GF ( 3 ) with length 3 m − 1 and dimension 3 m − 1 − 2 m according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied. [ABSTRACT FROM AUTHOR]

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three nonzeros. Recently, the authors of [37] constructed a class of cyclic codes with arbitrary number of nonzeros, and computed the weight distribution for several cases. In this paper, we determine the weight distribution for a new family of such codes. This is achieved by introducing certain new methods, such as the theory of Jacobi sums over finite fields and subtle treatment of some complicated combinatorial identities. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of q elements with q ≡ 3 ( mod 8 ) and p | q − 1 , where p is an odd prime. In this paper, we use matrix method to give all primitive idempotents in two rings F q [ x ] / 〈 x 4 p n − 1 〉 and F q [ x ] / 〈 x 8 p n − 1 〉 . Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length 4 p n and 8 p n over F q . [ABSTRACT FROM AUTHOR]

Let p be a prime number. In this paper, we study cyclic codes over the ring Z p [ u , v ] / 〈 u 2 , v 2 , u v − v u 〉 . We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes. We obtain all except one ternary optimal code of length 12 as the Gray image of the cyclic codes over the ring Z p [ u , v ] / 〈 u 2 , v 2 , u v − v u 〉 . [ABSTRACT FROM AUTHOR]

For two positive integers m and k , let C e be a class of cyclic code of length 2 m − 1 over F 2 with three nonzeros γ − 1 , γ − ( 2 k + 1 ) and γ − ( 2 e k + 1 ) for e = 2 or 3, where γ is a primitive element of F 2 m . When m gcd ⁡ ( m , k ) is odd, Kasami in 1971 determined the weight distributions of cyclic codes C 2 and C 3 , which is the same as that of the dual of the triple error-correcting BCH code. This paper obtains the weight distributions of C 2 and C 3 for the case of m gcd ⁡ ( m , k ) being even. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements and m 1 , m 2 two distinct positive integers such that gcd ⁡ ( m 1 , m 2 ) = d . Suppose that α 1 and α 2 are two primitive elements of F q m 1 and F q m 2 , respectively. Let n = ( q m 1 − 1 ) ( q m 2 − 1 ) / ( q d − 1 ) and T i denote the trace function from F q m i to F q for i = 1 , 2 . We define a cyclic code C ( q , m 1 , m 2 ) = { c ( a , b ) : a ∈ F q m 1 , b ∈ F q m 2 } , where c ( a , b ) = ( T 1 ( a α 1 0 ) + T 2 ( b α 2 0 ) , T 1 ( a α 1 1 ) + T 2 ( b α 2 1 ) , … , T 1 ( a α 1 n − 1 ) + T 2 ( b α 2 n − 1 ) ) . In this paper, we use Gauss sums to investigate the weight distribution of C ( q , m 1 , m 2 ) and prove that it has at most four nonzero weights if d = 2 and gcd ⁡ ( m 1 − m 2 2 , q − 1 ) = 1 . Furthermore, we get a class of three-weight cyclic codes if | m 1 − m 2 | = 2 . Some optimal or nearly optimal cyclic codes are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic self-orthogonal codes over Z 2 m for any odd length. We give the generator polynomials of cyclic self-orthogonal codes over Z 2 m . By using these generator polynomials, we obtain a sufficient and necessary condition for the existence of nontrivial cyclic self-orthogonal codes over Z 2 m and determine the number of such codes for a given odd length. Some results about cyclic self-dual codes over Z 2 m are included. [ABSTRACT FROM AUTHOR]

A lot of work has already been done in one- and two-weight irreducible cyclic codes. In particular, in the remarkable work of Schmidt and White [8] , all one- and two-weight irreducible cyclic codes were characterized, and it was conjectured that all of them are either subfield codes, or semiprimitive two-weight codes, or they belong to an exceptional set of eleven codes. In addition, the authors give, indirectly, a characterization of all semiprimitive two-weight irreducible cyclic codes over any finite field. The purpose of this work is to stress the importance of such characterization, and also, to present some remarks about one- and two-weight irreducible cyclic codes. This characterization and our remarks are important because through them we show that most of the recent results in one- and two-weight irreducible cyclic codes are, in one way or another, consequences of the work of Schmidt and White. For example, we will show that all new results regarding the two-weight irreducible cyclic codes, recently presented by C. Ding and J. Yang in [4] , can be viewed as mere instances of such characterization. Moreover, through this characterization we generalize the results about projective two-weight irreducible cyclic codes that were presented by J. Wolfmann in [11] . Our results will also be useful to clarify that the family of two-weight irreducible cyclic codes presented by Rao and Pinnawala, in [7] , is already included in the paper of Schmidt and White through an equivalence notion, and therefore, a major revision of the Schmidt and White conjecture is not needed. [ABSTRACT FROM AUTHOR]

We study the hulls of cyclic and negacyclic codes of length n over a finite field F q with respect to the Euclidean and Hermitian inner products. Based on the characterization of their generator polynomials, the dimensions of the hulls of cyclic and negacyclic codes of length n over F q are determined. The enumerations of cyclic codes and negacyclic codes of length n over F q having hulls of a given dimension are established as well. As special cases, known results concerning complementary dual and self-dual cyclic codes can be viewed as corollaries of results in this paper. [ABSTRACT FROM AUTHOR]

Suppose that l v | | q − 1 , where l is a prime and v is a positive integer, and 4 | q − 1 if l = 2 . In this paper, we give the weight distributions of all cyclic codes of length l m whose duals have at most two zeros and the weight distribution of a special code of length l m whose dual has three zeros. [ABSTRACT FROM AUTHOR]

Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first family of cyclic codes has parameters [3m-1,3m-1-2m,4] and contains a class of conjectured cyclic codes and several new classes of optimal cyclic codes. The second family of cyclic codes has parameters [3m-1,3m-2-2m,5] and contains a number of classes of cyclic codes that are obtained from perfect nonlinear functions over F3m, where m>1 and is a positive integer. [ABSTRACT FROM AUTHOR]

This note gives a counterexample of Theorem 20 in the paper of Blackford (2013) [2] . The counterexample shows that [2, Theorem 20] is incorrect. Furthermore, we provide corrections to the above result. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements and n = l 1 m 1 l 2 m 2 , m 1 ≥ 1 , m 2 ≥ 1 , where l 1 , l 2 are distinct primes and l 1 l 2 | q − 1 . In this paper, we give all irreducible factors of x l 1 m 1 l 2 m 2 − 1 over F q and all primitive idempotents in a ring F q [ x ] / 〈 x l 1 m 1 l 2 m 2 − 1 〉 . Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length l 1 m 1 l 2 m 2 over F q . [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and m , k be positive integers with m ≥ 3 k . Let C 1 and C 2 be cyclic codes over F p with parity-check polynomials h 2 ( x ) h 3 ( x ) and h 1 ( x ) h 2 ( x ) h 3 ( x ) , respectively, where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of γ − 1 , γ − ( p k + 1 ) and γ − ( p 3 k + 1 ) over F p , respectively, for a primitive element γ of F p m . Recently, Zeng et al. (2010) obtained the weight distribution of C 2 for m gcd ( m , k ) being odd. In this paper, we determine the weight distribution of C 1 , and the weight distribution of C 2 for the case that m gcd ( m , k ) is even. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in finding an algebraic structure of conjucyclic codes of length n over the finite field F 4. We show that conjucyclic codes of length n over F 4 are related to binary cyclic codes of length 2 n and show that there is a canonical bijective correspondence between the two sets. We illustrate how the factorization of the polynomial x 2 n + 1 plays a critical role in each setting. Moreover, we construct the generator and parity check matrices of conjucyclic codes of length n over F 4. [ABSTRACT FROM AUTHOR]

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over F q with designed distance δ = a q m − 1 − 1 (resp. δ = a q m − 1 q − 1) for all 1 ≤ a ≤ q − 1 , where q is a prime power and m > 1 is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16] , and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range a q m − 1 q − 1 , a q m − 1 q − 1 + T for 0 ≤ a ≤ q − 2 , where T = q m + 1 2 − 1 if m is odd, and T = 2 q m 2 − 1 if m is even. [ABSTRACT FROM AUTHOR]

Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over F 3 m . [ABSTRACT FROM AUTHOR]

Locally repairable codes with locality r (r -LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then (r , δ) -locally repairable codes ((r , δ) -LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r -LRC can be viewed as an (r , 2) -LRC. An (r , δ) -LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct q -ary optimal (r , δ) -LRCs with length much larger than q. Surprisingly, Luo et al. [3] presented a construction of q -ary optimal r -LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of q) via cyclic codes. In this paper, inspired by the work of [3] , we firstly construct two classes of optimal cyclic (r , δ) -LRCs with unbounded lengths and minimum distances δ + 1 or δ + 2 , which generalize the results about the δ = 2 case given in [3]. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic (r , δ) -LRCs with unbounded length and larger minimum distance 2 δ. Furthermore, when δ = 3 , we give another class of optimal cyclic (r , 3) -LRCs with unbounded length and minimum distance 6. [ABSTRACT FROM AUTHOR]

The component-wise or Schur product C ⁎ C ′ of two linear error-correcting codes C and C ′ over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in C ′. When C = C ′ , we call the product the square of C and denote it C ⁎ 2. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several "constituent" codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known (u , u + v) -construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes). [ABSTRACT FROM AUTHOR]

Let F 2 m be a finite field of cardinality 2 m , R = F 2 m + u F 2 m (u 2 = 0) and s , n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R of length 2 s n and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and 2-quasi-cyclic codes over F 2 m of length 2 s + 1 n can be obtained from self-dual cyclic code over R of length 2 s n and by a Gray map preserving orthogonality and distances from R onto F 2 m 2. [ABSTRACT FROM AUTHOR]