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There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as E = ⟨ a , b ∣ 2 a = 2 b = 0 , a 2 = a , b 2 = b , a b = a , b a = b ⟩. We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose codewords all have even Hamming weight. All these codes can be represented as formally self-dual additive codes over F 4. The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E. [ABSTRACT FROM AUTHOR]

In this paper, few weights linear codes over the local ring R=Fp+uFp+vFp+uvFp, with u2=v2=0,uv=vu, are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to Fp4, we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and p≡3(mod4). Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented. [ABSTRACT FROM AUTHOR]