On some special classes of linear and additive codes over finite commutative chain rings

Abstract

Self orthogonal codes, self-dual codes, and linear codes with complementary duals (LCD codes) constitute the three most important and well studied classes of linear codes. These codes have nice algebraic structures and are of great significance both from the practical and theoretical points of view. Self-orthogonal and self-dual codes have nice connections with the theory of designs and are useful in constructing secret-sharing schemes with nice access structures. LCD codes are useful in designing orthogonal direct-sum masking schemes, which protect sensitive information against side-channel attacks (SCA) and fault injection attacks (FIA). In the 1990s, it was shown that many binary non-linear codes can be viewed as Gray images of linear codes over the ring Z4 of integers modulo 4. Since then, much research has been devoted to studying self-orthogonal, self-dual, and LCD codes over finite commutative chain rings. In fact, the problem of the determination of enumeration formulae for self-orthogonal, self-dual, and LCD codes has attracted a great deal of attention, as these enumeration formulae are useful in classifying such codes up to equivalence. In this thesis, we obtain enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over finite commutative chain rings of odd characteristic. As special cases, one can obtain enumeration formulae for self-orthogonal and self-dual codes over quasi-Galois rings and Galois rings of odd characteristic. However, we observe that this enumeration technique can not be extended to count all self-orthogonal and self-dual codes over quasi-Galois rings and Galois rings of even characteristic. We modify this enumeration technique and provide explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over quasi-Galois and Galois rings of even characteristic. We also obtain explicit enumeration formulae for all LCD codes of an arbitrary length over finite commutative chain rings.