Construction of MDS self-dual codes over Galois rings

Abstract

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 [20]. 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(3 2 , 2), GR(3 3 , 2) and GR(3 4 , 2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5 2 , 2), GR(5 3 , 2) and GR(7 2 , 2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11 2 , 2) we have MDS self-dual codes of lengths up to 12.

Self-dual codes over commutative Frobenius rings

“…Proof. Its proof is a direct generalization of the corresponding proof in [16]. 2 A partial converse of the above theorem works as follows.…”

“…Let R be a finite Frobenius ring.If there is a c ∈ R such that c 2 + 1 = 0, then any self-dual codẽ C over R with free rank (C) 2, even length n 4, and minimum Hamming weight d > 2, can be constructed from some self-dual code C over R of length n − 2 by the construction method in Theorem 5.2.Proof. Its proof is a direct generalization of the corresponding proof in[16]. 2 Lemma 5.4.…”

“…A general construction, called the building-up construction, for self-dual codes over finite fields [15,1,4], Galois rings [16], and chain rings [7] was considered when the code length is even. In this section, we show that the same construction is also valid over a finite commutative Frobenius ring as well as over a finite local ring.…”

MASS FORMULA OF SELF-DUAL CODES OVER GALOIS RINGS GR(p2, 2)

“…. , k e−1 , k e which are nonnegative integers adding to n [7]. A code which have a generator matrix with this standard form is said to be of type (1) …”

Construction of extremal self-dual codes over $${\mathbb {Z}}_{8}$$ Z 8 and $${\mathbb {Z}}_{16}$$ Z 16

“…There has been active development on self-dual codes over finite fields or finite rings (for instance, [2,3,6,9,[12][13][14]19]). Recently, Dougherty et al [5] classified self-dual codes over Z 8 of lengths up to 8, and they [7] studied the lifted codes over finite chain rings and gave some results about the existence of self-dual codes over Z p r with p = 2.…”