Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

Projective two-weight irreducible cyclic and constacyclic codes

After several remarks on two-weight irreducible cyclic codes, we introduce a family of projective two-weight cyclic codes and a family of projective two-weight constacyclic codes and we discuss the existence of such codes.     

The weight distributions of irreducible cyclic codes of length

Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2^m over F_q are determined explicitly.     

Minimal cyclic codes of length pq

Explicit expressions for all the 3n+2 primitive idempotents in the ring R_pⁿq=GF(ℓ)[x]/(x^pnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pⁿ and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pⁿq over GF(ℓ) are also discussed.     

A class of minimal cyclic codes over finite fields

Let ${\mathbb{F}}_q$ be a finite field of odd order $q$ and $n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where $a,a_1,a_2$ are positive integers, $p_1,p_2$ are distinct odd primes and $4p_1{p}_2|q?1$. In this paper, we study the irreducible factorization of $x^n?1$ over ${\mathbb{F}}_q$ and all primitive idempotents in the ring ${\mathbb{F}}_q\left[x\right]∕〈x^n?1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length $n$ over ${\mathbb{F}}_q$.     

On circulant self-dual codes over small fields

We construct self-dual codes over small fields with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.      

2n Bordered constructions of self-dual codes from group rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes of length 64 and 68, constructing 30 new extremal self-dual codes of length 68.     

New classes of 2-weight cyclic codes

We introduce new classes of 2-weight cyclic codes which are direct sums of 1-weight irreducible cyclic codes      

Cardinal rank metric codes over Galois rings

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.     

An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings ${\mathbb{F}}_2+u{\mathbb{F}}_2$ and ${\mathbb{F}}_4+u{\mathbb{F}}_4$; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with $\gamma =5$ in $W_{68,2}$, which is the first instance of such a $\gamma$ value in the literature.     

Mass formula for self-dual codes over Z

We give a mass formula for self-dual codes over Z_p², where p is an odd prime. Using the mass formula, we classify such codes of lengths up to n=8 over the ring Z₉, n=7 over Z₂₅ and n=6 over Z₄₉.     

The nonexistence of near-extremal formally self-dual codes

A code is called formally self-dual if and have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over , and . These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally self-dual (f.s.d.) codes. With Zhang’s systematic approach, we determine possible lengths for which the four types of near-extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8|n was dealt with by Han and Lee.      

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.