Hamming code

Commutative ring

Algorithm

Mathematical physics

Discrete mathematics

Commutative property

Statistics

Local ring

Decoding methods

Organic chemistry

Hamming distance

Mathematics

Invariant (physics)

Block code

Combinatorics

Hamming weight

Chemistry

Linear code

Ring (chemistry)

Journal of Algebra and Its Applications

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

Type IV codes over a non-unital ring