Quasi type IV codes over a non-unital ring

Abstract

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations asWe give a natural map between linear codes over I and additive codes over 4 , that allows for efficient computations. We study the algebraic structure of linear codes over this non-unital local ring, their generator and parity-check matrices. A canonical form for these matrices is given in the case of so-called nice codes. By analogy with ℤ 4 -codes, we define residue and torsion codes attached to a linear I-code. We introduce the notion of quasi self-dual codes (QSD) over I, and Type IV I-codes, that is, QSD codes all codewords of which have even Hamming weight. This is the natural analogue of Type IV codes over the field 4 . Further, we define quasi Type IV codes over I as those QSD codes with an even torsion code. We give a mass formula for QSD codes, and another for quasi Type IV codes, and classify both types of codes, up to coordinate permutation equivalence, in short lengths.

The build-up construction over a commutative non-unital ring

“…the residue code defined by res(C) = {α(y) | y ∈ C}, 2. the torsion code defined by tor(C) = {x ∈ F n 2 | bx ∈ C}. It is easy to check that res(C) ⊆ tor(C) [3]. It is traditional to denote by k 1 the dimension of the residue code and k 1 + k 2 that of the torsion code.…”

“…Following a terminology introduced in [3], we shall call a QSD code with an even torsion code quasi Type IV (QTIV). Every Type IV code is quasi Type IV but not conversely as the next example shows.…”

“…This method was used successfully over a non-unital non commutative ring in [1]. In this article, we adapt this method to generate quasi self-dual (QSD) codes over the ring I, a non-unital, commutative ring of order 4 [3]. QSD codes are defined in that reference as self-orthogonal codes of length n, and size 2 n .…”

“…Of special interest is the subclass of Type IV codes, namely QSD codes with all weights even, which exists for several rings of order four [10,2]. A relaxation of that concept that is special to the ring I is that of quasi Type IV codes (QTIV), that is to say QSD codes with an even torsion code [3]. While the build up constructions do not seem to preserve the class of Type IV codes, they are shown here to preserve the wider class of QTIV codes.…”

DNA codes over two noncommutative rings of order four

“…In fact, there are some linear codes over rings of order four which are neither GF p4q nor Z 4 [2], [3], [4]. In particular, we construct DNA codes satisfying constraints over the two noncommutative rings E and F in the notation of [7].…”