In this work, we present a new construction method for reversible codes. We employ composite matrices derived from group rings and show how to construct these matrices so that they are also reversible. Also in this work, we give an algorithm for calculating conflict free DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement, the GC-content constraints with each DNA codeword being free from reverse complement sub-strings. By employing our construction method for reversible codes and our algorithm, we construct a number of DNA codes that satisfy the above constraints. Many of the codes we obtain have better parameters than some known DNA codes and many have parameters that are new to the literature.

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths$n \leq 10$. The obtained codes are either optimal or near optimal according to Bierbraueret al+. (2009).

Abstract In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $$\gamma =7,8$$ γ = 7 , 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction

A subset of a vector space$\mathbb {F}_{q}^{n}$is additive if it is a linear space over the field$\mathbb {F}_{p}$, where$q=p^{e}$,$p$prime, and$e>1$. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case$e=2$, it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

A subset of a vector space Fqnis K-additive if it is a linear space over the subfield K ⊆ Fq. Let q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are Fp-additive are established. For specific ranks and dimensions of the kernel within these bounds, Fp-additive GH codes are constructed. Moreover, for the case e = 2, it is shown that the given bounds are tight and it is possible to construct an Fp-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

A ℤ2ℤ4-additive code C ⊆ ℤ2α× ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2coordinates and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Φ(C) be the binary Gray map image of C. We study the rank and the dimension of the kernel of a ℤ2ℤ4-additive cyclic code C, that is, the dimensions of the binary linear codes (Φ(C)) and ker (Φ(C)). We give upper and lower bounds for these parameters. It is known that the codes (Φ(C)) and ker (Φ(C)) are binary images of ℤ2ℤ4-additive codes that we denote by R(C) and K(C), respectively. Moreover, we show that R(C) and K(C) are also cyclic and determine the generator polynomials of these codes in terms of the generator polynomials of the code C.

A 762764-additive code C ⊆ ℤ2α× ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of ℤ2ℤ4-additive cyclic codes. We determine all ℤ2ℤ4-additive cyclic codes with odd β whose Gray images are linear binary codes. In this case, it is shown that such binary codes are permutation equivalent (by the Nechaev permutation) to ℤ2-double cyclic codes. Finally, the generator polynomials of these binary codes are given.

The ranks and kernels of generalized Hadamard matrices are studied. It is proved that any generalized Hadamard matrix H(q, λ) over Fq, q > 3, or q = 3 and gcd(3, λ) ≠ 1, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed.

Following [5], given a Z2Z4-additive self-dual code, one can easily extend this code and generate an extended Z2Z4-additive self-dual code with greater length. In this communication we study these constructions and check if properties like separability and code Type are retained or not.

We describe recent results for codes over Z2×Z4giving their connection to binary codes via a natural Gray map. We study Z2Z4self-dual codes and we state the major results concerning these codes. We state several open questions and discuss possible avenues of research.

Secret sharing is an important topic in cryptography and has applications in information security. We use self-dual codes to construct secret-sharing schemes. We use combinatorial properties and invariant theory to understand the access structure of these secret-sharing schemes. We describe two techniques to determine the access structure of the scheme, the first arising from design properties in codes and the second from the Jacobi weight enumerator, and invariant theory.

We describe various constructions of unimodular lattices from codes over finite rings and modular forms constructed from those lattices.

We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 3/spl lscr/ as codes of length /spl lscr/ over the ring F/sub 2//spl times/F/sub 4/. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New extremal binary codes in lengths /spl les/ 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72,36,16] Type II code are derived.

We define the complete joint weight enumerator in genus g for codes over /spl Zopf//sub 2k/ and use it to study self-dual codes and their shadows. These weight enumerators are related to the theta series of the associated lattices and Siegel and Jacobi forms are formed from these series.

In this correspondence, we study bounds on the Euclidean, Hamming, Lee, and Bachoc weights of codes over rings of order 4 similar to the Singleton bound and investigate the relationship between these bounds. Moreover, we give some characterizations of the codes meeting these bounds.

In this correspondence, we study maximum distance with respect to rank (MDR) codes over the ring Z/sub k/. We generalize the construction of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon codes and apply the generalized Chinese remainder theorem to construct codes.

We study self-dual codes over the ring Z/sub 2k/ of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over Z/sub 2k/ which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extremal Type II code over Z/sub 6/ of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over Z/sub 2k/ are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over Z/sub 2k/ and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over Z/sub 2k/ are also introduced.

We study Type IV self-dual codes over the commutative rings of order 4. Gleason-type theorems of Type IV codes and their shadow codes are investigated. A mass formula of Type IV codes over these rings are given. We give a classification of Type TV codes over Z/sub 4/ and F2+uF/sub 2/ for reasonable lengths. We also construct a number of optimal Type TV codes.

The alphabet F/sub 2/+uF/sub 2/ is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F/sub 2/+uF/sub 2/ codes with Lee weights a multiple of 4 are called Type II. They give even unimodular Gaussian lattices by Construction A, while Type I codes yield unimodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of the Type I codes yields bounds on the highest minimum Hamming and Lee weights.

We give two computational results on binary self-dual codes. A number of extremal singly-even self-dual [68, 34, 12] codes with weight enumerators not known to exist, are constructed. For k/spl les/10, the relationship between extremal doubly-even self-dual codes of length 24 k, and certain singly-even self-dual codes of length 24 k-2 is presented.

In this correspondence, we investigate binary extremal self-dual codes. Numerous extremal self-dual codes and interesting self-dual codes with minimum weight d=14 and 16 are constructed. In particular, the first extremal Type I [86,43,16] code and new extremal self-dual codes with weight enumerators which were not previously known to exist for lengths 40,50,52 and 54 are constructed. We also determine the possible weight enumerators for extremal Type I codes of lengths 66-100.

The technique of using shadow codes to build larger self-dual codes is extended to codes over arbitrary fields. It is shown how to build the codes and how to determine the new weight enumerator as well. For codes over fields equipped with a square root of -1 and not of characteristic 2, a self-dual code of length n+2 can be built from a self-dual code of length n; for codes over a field without a square root of -1 and not of characteristic 2 a self-dual code of length n+4 is built from a self-dual code of length n; and for codes over fields of characteristic 2 the length of the new self-dual code depends on the presence of the all-one vector in the subcode chosen. In certain cases using the subcode of vectors orthogonal to the all-one vector, the new weight enumerator can be calculated directly from the original weight enumerator. Specific examples of the technique are illustrated for codes over F/sub 3/, F/sub 4/, and F/sub 5/.>