Abstract
There exist two semi-local rings of order 6 without identity for the multiplication. We classify up to coordinate permutation self-orthogonal codes of length n and size 6n/2 over these rings (called here quasi self-dual codes or QSD) till the length n = 8. To any such code is attached canonically a ℤ6-code, which, when self-dual, produces an unimodular lattice by Construction A.
Highlights
Codes over the ring of order six Z6 have received some attention in the past due to their connection to euclidean lattices [11,13]
There is a non multiplicative analogue of the Chinese Remainder Theorem (CRT) that allows to attach to any code over such a ring the ordered pair of a binary code and a ternary code
If the code is quasi self-dual (QSD) that is self-orthogonal of length n and size 6n/2, it can be shown that one of the two codes is self-dual and the other is a rate one-half code
Summary
Codes over the ring of order six Z6 have received some attention in the past due to their connection to euclidean lattices [11,13]. In a series of recent papers, the authors have studied self-orthogonal codes over nonunital rings of order 4 [1, 2, 4, 3]. We study self-orthogonal codes over two non-unital rings of order 6, denoted here by H32 and H23. Both rings are semilocal with two maximal ideals of size two and three. If the code is quasi self-dual (QSD) that is self-orthogonal of length n and size 6n/2, it can be shown that one of the two codes is self-dual and the other is a rate one-half code Forgetting their multiplicative structure, we can regard the codes over either of these two non-unital rings as additive codes over Z6, or, equivalently Z6-linear codes.
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