The number of self-dual codes over $${Z_{p^3}}$$

Let p be a prime number. In this paper, we consider codes over the ring of integers modulo p ³ and give a characterization of self-duality. This leads to a construction of self-dual codes and a mass formula, which counts the number of such codes over .      

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.      

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.      

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let $R$ be the Galois ring $GR(p^k,s)$ of characteristic $p^k$ and cardinality $p^{sk}$. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length $n$ over $R$, and a $p$-adic integer ring with $gcd(p,n)=1$. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length $r{l}^m$ over $R$, where $r,l,$ and $t$ are three primes with $2?l$, $r|(q^t?1)$, $l^v\parallel (q^t?1)$ and $gcd(rl,q(q?1))=1$. Finally, as applications, weight distributions of all irreducible cyclic codes for $t=2$ and generator polynomials of self-dual cyclic codes of length $l^m$ and $r{l}^m$ over $R$ are given.     

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.      

Binary self-dual codes with automorphisms of order 23

The only example of a binary doubly-even self-dual [120,60,20] code was found in 2005 by Gaborit et al. (IEEE Trans Inform theory 51, 402–407 2005). In this work we present 25 new binary doubly-even self-dual [120,60,20] codes having an automorphism of order 23. Moreover we list 7 self-dual [116,58,18] codes, 30 singly-even self-dual [96,48,16] codes and 20 extremal self-dual [92,46,16] codes. All codes are new and present different weight enumerators.      

New constructions of optimal self-dual binary codes of length 54

The quaternary Hermitian self-dual [18,9,6]₄ codes are classified and used to construct new binary self-dual [54,27,10]₂ codes. All self-dual [54,27,10]₂ codes obtained have automorphisms of order 3, and six of their weight enumerators have not been previously encountered.      

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.     

On the classification of all self-dual additive codes over GF(4) of length up to 12

We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complementation and graph isomorphism. We use these facts to classify all codes of length up to 12, where previously only all codes of length up to 9 were known. We also classify all extremal Type II codes of length 14. Finally, we find that the smallest Type I and Type II codes with trivial automorphism group have length 9 and 12, respectively.     

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.     

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.     

Classification of cyclic codes over a non-Galois chain ring Zp[u]/〈u3〉

We explicitly determine generators of cyclic codes over a non-Galois finite chain ring ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$, where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$ and four types of non-principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$, which are associated with cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$. We then obtain a mass formula for cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$.