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DNA-based storage has been a promising technique of data storage, due to its high density and long duration. During synthesizing and sequencing of DNA storage, edit errors including insertions, deletions and substitutions are introduced inevitably. An effective way to reduce the error probability is to limit the content of G and C in DNA sequences to around 50%, which is called GC-balanced. To deal with edit errors, DNA sequences are also expected to have error-correcting capabilities. In this paper, GC globally balanced and GC locally balanced error-correcting codes are explicitly constructed, respectively. Inspired by repetition codes, the proposed codes are able to correct multiple edit errors. Furthermore, an efficient decoding algorithm applied for both codes is derived when only one kind of edit error occur. [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]

Abstract Binary asymmetric error-correcting codes (AECC) play an important role in communication systems modeled by the binary symmetric channel (Z-channel). Varshamov proposed a general construction of binary AECC via V h -sequences and obtained many good binary AECC. In this paper, we give a general construction of V h -sequences in ray class groups of algebraic function fields. Our construction induces new lower bounds for binary AECC which improves the results in [18]. [ABSTRACT FROM AUTHOR]

The construction of quantum maximum distance separable (abbreviated to MDS) error-correcting codes has become one of the major concerns in quantum coding theory. In this paper, we further generalize the approach developed in the previous paper, and construct several new classes of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. By employing these classical MDS codes, we obtain several classes of quantum MDS codes with large minimum distance. It turns out that our quantum MDS codes exhibited here have less constraints on the selection of code length and some of them have not been constructed before, and in some cases, have larger minimum distance than previous literature. Meanwhile, about half of the distance parameters of our codes are greater than q 2 + 1. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extension, we focus our study on MDS linear codes. It is well-known that an MDS linear code with minimum distance 2 ℓ + 1 has an ℓ -error-correcting pair if and only if it is a generalized Reed-Solomon code. In this paper, we show that for an MDS linear code C with minimal distance 2 ℓ + 2 , if it has an ℓ -error-correcting pair, then the parameters of the pair are three cases. For one case, we give a necessary condition that C is a generalized Reed-Solomon code, and then give some counterexamples that C is a non-generalized Reed-Solomon code for the other two cases. [ABSTRACT FROM AUTHOR]

In this paper, we characterize hulls of double cyclic codes over Z 2. Firstly, we give the generators of hulls of double cyclic codes under the separable case and the non-separable case with coprime odd block lengths. Then, using the concept of a generating function in combinatorics, the solution of dimensions of hulls of double cyclic codes over Z 2 is given. Furthermore, the enumeration of double cyclic codes over Z 2 with hulls of a fixed dimension is determined. As an application, some good entanglement-assisted quantum error-correcting codes (EAQECCs) are obtained by the hulls of non-separable double cyclic codes. [ABSTRACT FROM AUTHOR]

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over F q m are GRS codes (resp., extended GRS codes) when m = 1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs. [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 of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the generalized Hamming weights of three classes of linear codes constructed through defining sets and determine them partly for some cases. Particularly, in the semiprimitive case we solve a problem left in Yang et al. (2015) [30] . [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]

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace [18]. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in [18] and with excellent parameters. Namely, we are able to provide new binary records according to [21] and non-binary codes improving the ones available in the literature. [ABSTRACT FROM AUTHOR]

Goppa codes were defined by Valery D. Goppa in 1970. In 1978, Robert J. McEliece used this family of error-correcting codes in his cryptosystem, which has gained popularity in the last decade due to its resistance to attacks from quantum computers. In this paper, we present Goppa codes over the p -adic integers and integers modulo p e. This allows the creation of chains of Goppa codes over different rings. We show some of their properties, such as parity-check matrices and minimum distance, and suggest their cryptographic application, following McEliece's scheme. [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]

Several bounds on the size of ( n + l , M , d , ( m , 0 ) ) q codes in attenuated space over finite fields are provided in this paper. Then, we prove that codes in attenuated space attain the Wang–Xing–Safavi-Naini bound if and only if they are certain Steiner structures. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding of an orthogonal Grassmannian . In particular, we determine some of the parameters of the codes arising from the projective system determined by . We also study special sets of points of which are met by any line of in at most 2 points and we show that their image under the Grassmann embedding is a projective cap. [Copyright &y& Elsevier]

Abstract: Castle codes are algebraic geometry one-point codes on Castle curves. This family contains some of the most important AG codes among those studied in the literature to date. The minimum distance of these codes can be bounded by using the order-like bound , which is known to be equivalent to the classical order bound when both can be applied. In this paper we compute for some Castle codes, including those related to semigroups generated by two elements and telescopic semigroups. In particular we compute the bound in full for Suzuki codes. [Copyright &y& Elsevier]

In 1985, Gabidulin introduced the rank metric in coding theory over finite fields, and used this kind of codes in a McEliece cryptosystem, six years later. In this paper, we consider rank metric codes over Galois rings. We propose a suitable metric for codes over such rings, and show its main properties. With this metric, we define Gabidulin codes over Galois rings, propose an efficient decoding algorithm for them, and hint their cryptographic application. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we construct some algebraic geometric error-correcting codes on surfaces whose Néron–Severi group has low rank. If the Néron–Severi group is generated by an effective divisor, the intersection of this surface with an irreducible surface of lower degree will be an irreducible curve, and this makes possible the construction of codes with good parameters. Such surfaces are not easy to find, but we are able to find surfaces with low rank, and those will give us good codes too. [Copyright &y& Elsevier]

Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability [6] in the classical setting. This interest also translates to other metric setting, including the insertion and deletion (insdel for short) setting which is used to model synchronization errors caused by positional information loss in communication systems. Such interest is supported by the construction of a deletion correcting algorithm of insdel Reed-Solomon code in [22] which is based on the Guruswami-Sudan decoding algorithm [6]. Nevertheless, there have been few studies [3] on the insdel error-correcting capability of Reed-Solomon codes. In this paper, we discuss a criterion for a 2-dimensional insdel Reed-Solomon codes to have optimal asymptotic error-correcting capabilities, which are up to their respective lengths. Then we provide explicit constructions of 2-dimensional insdel Reed-Solomon codes that satisfy the established criteria. The family of such constructed codes can then be shown to extend the family of codes with asymptotic error-correcting capability reaching their respective lengths provided in [3, Theorem 2] which provide larger error-correcting capability compared to those defined in [25]. [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]