There is a special local ring $E$ $E$ of order $4,$ $4,$ without identity for the multiplication, defined by $E = ⟨ a, b ∣ ∣ 2 a = 2 b = 0, a^{2} = a, b^{2} = b, a b = a, b a = b ⟩ .$ $E = 〈 a, b | 2 a = 2 b = 0, a^{2} = a, b^{2} = b, a b = a, b a = b 〉 .$ 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 $E,$ $E,$ 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.

Communicated by S. K. Jain

Keywords:

AMSC: 94B05, 16A10

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