Search

Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on m $m$‐by‐n $n$ matrices over a finite field Fq ${{\mathbb{F}}}_{q}$. They are diameter perfect and have the cardinality qm(n−d+1) ${q}^{m(n-d+1)}$ if m≥n $m\ge n$. We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in m $m$ if the other parameters (n,q $n,\,q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. [ABSTRACT FROM AUTHOR]

We study codes with parameters of $q$ -ary shortened Hamming codes, i.e., $(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$ -fold packing with minimum distance 2. Secondly, for every admissible length starting from $n=20$ , we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code. [ABSTRACT FROM AUTHOR]

In this paper, we give a new method to answer a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the equation x d + (x + 1) d = b in F q 4 , where n is a positive integer, q = 2 n and d = q 3 + q 2 + q − 1. In particular, we directly determine the differential spectrum of this power function x d using a more simple and direct method compared with those in the literature. [ABSTRACT FROM AUTHOR]

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7]. [ABSTRACT FROM AUTHOR]

We prove that the covering radius of the Melas code $M(m,q)$ of length $n=q^{m}-1$ over $\mathbb {F}_{q}$ is 2 if $q > 3$. We also prove that the covering radius of $M(m,3)$ is 3 is $m \ge 3$ , the covering radius of $M(2,3)$ is 4, and the covering radii of $M(1,2)$ and $M(1,3)$ are 1. [ABSTRACT FROM AUTHOR]

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over Z p m , for p a prime, and more generally over chain rings of depth m, and with a residue field of size q, a prime power. In the case of p = m = 2 , strong necessary conditions (diophantine equations) on the weight distribution are derived, leading to a partial classification in modest length. Infinite families of examples are built from Kerdock and generalized Teichmüller codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear. [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]

The symbol-pair read channel was first proposed by Cassuto and Blaum. Later, Yaakobi et al. generalized it to the b -symbol read channel. It is motivated by the limitations of the reading process in high density data storage systems. One main task in b -symbol coding theory is to determine the b -symbol weight hierarchy of codes. In this paper, we study the b -symbol weight hierarchy of the Kasami codes, which are well known for their applications to construct sequences with optimal correlation magnitudes. The complete symbol-pair weight distribution of the Kasami codes is determined. [ABSTRACT FROM AUTHOR]

We investigate fat projective linear codes over ${\mathbb Z}_{p^{m}}$ , $m\geqslant 2$ , with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator matrix of the code determine a cap in the projective Hjelmslev geometry $\mathop {\mathrm {PHG}}\nolimits (k-1, {\mathbb Z}_{p^{m}})$. This generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982). The proof relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry. [ABSTRACT FROM AUTHOR]

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the $\mathbb {Z}_{2} \mathbb {Z}_{4}$ -additive quasi-cyclic codes. [ABSTRACT FROM AUTHOR]

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in Q-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given. [ABSTRACT FROM AUTHOR]

We study the k-Galois linear complementary dual (LCD) codes over the finite chain ring R = Fq + uFq with u2 = 0, where q = pe and p is a prime number. We give a sufficient condition on the generator matrix for the existence of k-Galois LCD codes over R. Finally, we show that a linear code over R (for k = 0, q > 3) is equivalent to a Euclidean LCD code, and a linear code over R (for 0, (pe−k + 1) (pe−1) and (pe−1)/(pe−k + 1) > 1) is equivalent to a k-Galois LCD code. [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]

Recently, several infinite families of minimal and optimal linear codes are obtained from simplicial complexes or down-sets by Hyun and Wu et al. Motivated by their work, we give two new constructions of codes over the ring F p + u F p , u 2 = u by using down-sets. We compute the Lee weight distributions of two classes of codes when the down-sets are generated by a single maximal element. Moreover, by using the Gray map, we obtain two classes of codes, which are minimal and distance optimal. [ABSTRACT FROM AUTHOR]

If S is a transitive metric space, then | C | ⋅ | A | ≤ | S | for any distance- d code C and a set A , "anticode", of diameter less than d. For every Steiner S (t , k , n) system S , we show the existence of a q -ary constant-weight code C of length n , weight k (or n − k), and distance d = 2 k − t + 1 (respectively, d = n − t + 1) and an anticode A of diameter d − 1 such that the pair (C , A) attains the code–anticode bound and the supports of the codewords of C are the blocks of S (respectively, the complements of the blocks of S). We study the problem of estimating the minimum value of q for which such a code exists, and find that minimum for small values of t. [ABSTRACT FROM AUTHOR]

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function $\Gamma _{Q}(n,\ell,k,q)$ , that is defined to be the largest number of nonzero weights a quasi-cyclic code of index $\gcd (\ell,n)$ , length $n$ and dimension $k$ over $\mathbb F_{q}$ can have, and connect it to similar functions related to linear and cyclic codes. We provide several upper and lower bounds on this function, using different techniques and studying its asymptotic behavior. Moreover, we determine the smallest index for which a $q$ -ary Reed-Muller code is quasi-cyclic, a result of independent interest. [ABSTRACT FROM AUTHOR]

The dual of the Kasami code of length $q^{2}-1$ , with $q$ a power of 2, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_{q}$ with a Simplex code of length $q-1$. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related $q$ -ary MDS code are determined. New cyclic completely regular codes over finite fields a power of 2, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code. [ABSTRACT FROM AUTHOR]

In this paper, we investigate all repeated-root constacyclic codes and their duals of length k ℓ p s over F q in terms of generator polynomials, where ℓ is an odd prime different from p and k is an odd prime different from both ℓ and p such that k = 4 h + 1 for some prime h. As an application, the characterization and enumeration of all self-dual repeated-root cyclic codes of length 2 s k ℓ over F q are obtained. [ABSTRACT FROM AUTHOR]

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function $\Gamma (k,q)$ , that is defined as the largest number of nonzero weights a cyclic code of dimension $k$ over $\mathbb {F}_{q}$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or $q$ -ary, as well as in the $q$ -ary Hamming code are determined, two difficult results of independent interest. [ABSTRACT FROM AUTHOR]

The p‐rank of a Steiner triple system (STS) B is the dimension of the linear span of the set of characteristic vectors of blocks of B, over GF(p). We derive a formula for the number of different STSs of order v and given 2‐rank r2, r2

Modular strongly regular graphs have been introduced by Greaves et al. (Linear Algebra Appl 639:50–80, 2022). We show that a related class of isodual codes is asymptotically good. Equiangular tight frames over finite fields also introduced by the same authors in 2022 are shown here to connect with self-dual codes. We give several examples of equiangular tight frames over finite fields arising from self-dual codes. [ABSTRACT FROM AUTHOR]

In this paper, all of the conjectures in Aydin et al. (2022) [6] are settled: some proven correct, some proven correct with a modification, and some disproven, involving the equivalence and duality of polycyclic codes associated with trinomials. We also give methods to construct isodual and self-dual polycyclic codes, and study the self-orthogonal and dual-containing polycyclic codes over F 2. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Toeplitz codes of index t , which are a generalization of quasi-cyclic codes of index t. We show that such codes meet the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 1 t over F q , so they are asymptotically good for any t. We characterize some special classes of Toeplitz codes as LCD codes (resp. linear codes with one-dimensional hull or self-orthogonal codes). It is worth mentioning that we determine the hull dimension of the isometric image of a linear code. Some optimal or almost optimal binary linear codes are obtained as a result. Most importantly, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) from quasi-cyclic codes. In particular, we construct binary LCD [ 32 , 16 , 8 ] , [ 33 , 11 , 11 ] and [ 34 , 17 , 8 ] codes with improved parameters directly. [ABSTRACT FROM AUTHOR]

In this paper, several classes of three-weight codes and two-weight codes for the homogeneous metric over the chain ring R = F p + u F p + ⋯ + u k - 1 F p , with u k = 0 , are constructed that generalize the construction of Shi et al. (IEEE Commun. Lett. 20(12):2346–2349, 2016), which is the special case of p = k = 2. These codes are defined as trace codes. In some cases of their defining sets, they are abelian. Their homogeneous weight distributions are computed by using exponential sums. In particular, in the two-weight case, we give some conditions of optimality of their Gray images by using the Griesmer bound. Their dual homogeneous distance is also given. The codewords of these codes are shown to be minimal for inclusion of supports, a fact favorable to an application to secret sharing schemes. [ABSTRACT FROM AUTHOR]

The codes in the title are studied with respect to existence, enumeration and asymptotic performance. Their Gray images are shown to satisfy a modified Gilbert-Varshamov bound. Explicit counting formulas are derived. Examples of modest lengths are given where Gray images of LCD Z 4 -codes outperform the best known binary linear LCD codes. [ABSTRACT FROM AUTHOR]

In this paper, two different Gray-like maps from $Z_{p}^\alpha \times Z_{p^{k}}^\beta $ , where $p$ is prime, to $Z_{p}^{n}$ , $n={\alpha +\beta p^{k-1}}$ , denoted by $\varphi $ and $\Phi $ , respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_{p} Z_{p^{k}}$ -additive code, and $C^\bot $ is its dual, then the weight enumerators of the image $p$ -ary codes $\varphi (C)$ and $\Phi (C^\bot)$ are formally dual. This is a partial generalization of [D. S. Krotov, On $Z_{2^{k}}$ -dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532–1537], and the result is generalized to odd characteristic $p$ and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed $Z_{p} Z_{p^{2}}\ldots Z_{p^{k}}$ alphabet is given. [ABSTRACT FROM AUTHOR]

We study double circulant LCD codes over ℤ p 2 for all odd primes p , and self-dual double circulant codes over ℤ p 2 for primes p ≡ 1 (mod   4). We derive exact enumeration formulae, and asymptotic lower bounds on the minimum distance of the p -ary images of these codes by the classical Gray maps. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, code extension and projection over a self-dual basis are the main tools. Some optimal, almost MDS, and MDS self-dual codes over both small and large finite fields are constructed. Moreover, over fifty MDS codes with new parameters are constructed. Comparisons with classical constructions are made. [ABSTRACT FROM AUTHOR]

Four circulant codes form a special class of 2 -generator, index 4 , quasi-cyclic codes. Under some conditions on their generator matrix they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an infinite subclass of these codes satisfying a modified Gilbert–Varshamov bound. [ABSTRACT FROM AUTHOR]

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis q, we show that there are infinitely many pairs of primes the base q expansion of which differ in at most two digits. Likewise, for any fixed integer t, there are infinitely many pairs of primes, the first t digits of which are the same. In another direction, we show that, there is a constant c depending on q such that for infinitely many integers m there are at least c log log m primes which differ from m by at most one base q digit. [ABSTRACT FROM AUTHOR]

We construct two families of few-weight codes for the Lee weight over the ring $R_{k}$ based on two different defining sets. For the first defining set, taking the Gray map, we obtain an infinite family of binary two-weight codes which are in fact $2^{k}$ -fold replicated MacDonald codes. For the second defining set, we obtain two infinite families of few-weight codes. These few-weight codes can be used to implement secret-sharing schemes. [ABSTRACT FROM AUTHOR]

We study linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$ when $q$ is an odd prime power. When $q^{\unicode[STIX]{x1D6FF}}+1$ is divisible by $n$ , we obtain an exact count of linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$. For certain values of $n$ and $q$ and assuming Artin’s conjecture for primitive roots, we show that the relative distance of these codes satisfies a modified Gilbert–Varshamov bound. [ABSTRACT FROM AUTHOR]

In this paper, few weights linear codes over the local ring R=Fp+uFp+vFp+uvFp, with u2=v2=0,uv=vu, are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to Fp4, we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and p≡3(mod4). Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented. [ABSTRACT FROM AUTHOR]

We study codes with parameters of the ternary Hamming (n = (3 m − 1) / 2 , 3 n − m , 3) code, i.e., ternary 1-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary 1-perfect codes of rank n − m + 1 , count their number, and prove that all such codes can be obtained from each other by a sequence of two-coordinate switchings. We enumerate ternary 1-perfect codes of length 13 obtained by concatenation from codes of lengths 9 and 4; we find that there are 93241327 equivalence classes of such codes. [ABSTRACT FROM AUTHOR]

In the paper, we firstly study the algebraic structures of Z p Z p k -additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for the Z p Z p 2 -additive cyclic code whose Gray image is linear (not necessarily cyclic) over Z p is given. Moreover, as for some special families of cyclic codes over Z p 2 and Z p Z p 2 , the linearity of the Gray images is determined. [ABSTRACT FROM AUTHOR]

In this paper 1 , we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis. [ABSTRACT FROM AUTHOR]

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring F p + u F p + v F p + u v F p , where u 2 = 0 , v 2 = 0 , u v = v u . These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over F p . In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given. [ABSTRACT FROM AUTHOR]

We study a special class of quasi-cyclic codes, obtained from a cyclic code over an extension field of the alphabet field by taking its image on a basis. When the basis is equal to its dual, the dual code admits the same construction. We give some examples of self-dual codes and LCD codes obtained in this way. [ABSTRACT FROM AUTHOR]

We construct two new infinite families of trace codes of dimension 2m , over the ring \mathbb {F}_{p}+u\mathbb {F}_{p} , with u^{2}=u , when p is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear p -ary codes of respective lengths and 2(p^m-1)^2 . When m is singly even, the first family gives five-weight codes. When m is odd and p\equiv 3 \pmod {4} , the first family yields p$ -ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever $p=3$ and $m \ge 3$ , or $p\ge 5$ and $m\ge 4$ . Applications to secret sharing schemes are given. [ABSTRACT FROM PUBLISHER]

We first define a new Gray map from R = Z 4 + u Z 4 to Z 4 2 , where u 2 = 1 and study ( 1 + 2 u ) -constacyclic codes over R . Also of interest are some properties of ( 1 + 2 u ) -constacyclic codes over R . Considering their Z 4 images, we prove that the Gray images of ( 1 + 2 u ) -constacyclic codes of length n over R are cyclic codes of length 2 n over Z 4 . In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic R -codes published by Özen et al. [7] . [ABSTRACT FROM AUTHOR]

We construct an infinite family of two-Lee-weight codes over the ring F 2 + u F 2 + v F 2 + u v F 2 . These codes are defined as trace codes and have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Then, taking Gray mapping, we obtain an infinite family of abelian binary two-weight codes which are shown to be optimal by application of the Griesmer bound. Moreover, an application to secret sharing schemes is given. [ABSTRACT FROM AUTHOR]

Let $R=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}+\cdots +u^{d-1}\mathbb{F}_{p}$, where $u^{d}=u$ and $p$ is a prime with $d-1$ dividing $p-1$. A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{dk}$ over $R$ and the dual code $\mathscr{C}^{\bot }$ is established. [ABSTRACT FROM PUBLISHER]

In this paper, we solve the open problems raised in [8] and present some examples to illustrate the obtained results. Moreover, we work out the diophantine problem by Shi and Wang and then give the sufficient conditions for the nonexistence of two-Lee weight projective codes over \mathbb Z4 with type 4^k12^{k2} . [ABSTRACT FROM PUBLISHER]

We show that no more new distance-regular graphs in the tables of the book of (Brouwer, Cohen, Neumaier, 1989) can be produced by using the coset graph of additive completely regular codes over finite fields. [ABSTRACT FROM AUTHOR]

An additive cyclic code of length n over F 4 can be defined equivalently as an F 2 [ x ] / 〈 x n + 1 〉 -submodule of F 4 [ x ] / 〈 x n + 1 〉. In this paper we study additive cyclic and complementary dual codes of odd length over F 4 with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of these codes by their generators as binary cyclic codes. [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]

This paper is devoted to the construction of two-Lee weight codes overwith typebased on a distance-preserving Gray map fromto. The necessary conditions for a code overto have only two different nonzero weights are given. The existence of two-Lee weight projective codes is also presented. Finally, some examples are also presented to illustrate the algebraic structure of two-Lee weight codes over. [ABSTRACT FROM PUBLISHER]

In this paper, we construct new infinite families of codes over the ring R = F 2 + u F 2 + v F 2 + u v F 2 , u 2 = v 2 = 0 , u v = v u by employing simplicial complexes. When the simplicial complexes are all generated by a single maximal element, we compute the Lee weight distributions of these classes of codes. By the Gray map, we show that some classes of codes are minimal and some classes of these codes are distance optimal. [ABSTRACT FROM AUTHOR]