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. Besides this, we show that the class of -LCD codes over finite commutative chain rings is asymptotically good. We also show that every free linear code over a finite commutative chain ring is equivalent to a -LCD code when the residue field of the chainring is of order at least 5. We also explicitly determine all inequivalent -LCD codes of length n, rank k and Hamming distance d over a finite commutative chain ring when k 2 {1, n 1} and 1  d  n. We further study additive codes over finite commutative chain rings and their dual codes with respect to the ordinary trace bilinear form in the Galois additivity case. We derive necessary and sucient conditions under which an additive code over a finite commutative chain ring is (i) self-orthogonal, (ii) self-dual, and (iii) an additive code with complementary dual (or an ACD code). We further provide enumeration formulae for all additive self-orthogonal and self-dual codes of an arbitrary length over finite commutative chain rings in certain special cases. We also count all ACD codes of an arbitrary length over finite commutative chain rings. We further show that a free additive code over a finite commutative chain ring is a maximum distance separable code (or an MDS code) if and only if its Torsion code is an additive MDS code. This motivates us to introduce and study two new classes of additive codes over finite fields, viz. additive generalized Reed-Solomon (additive GRS) codes and additive generalized twisted Reed-Solomon (additive GTRS) codes, which are extensions of linear GRS codes and linear GTRS codes, respectively. Unlike linear GRS codes, we note that additive GRS codes are not MDS codes in general. We also identify several new classes of additive MDS and almost MDS codes within the families of additive GRS and GTRS codes. We also note that, unlike linear codes, the dual code of an additive MDS code need not be an additive MDS code. We identify several classes of additive MDS codes whose dual codes are also MDS within the families of additive GRS and GTRS codes. We provide constructions of additive MDS self-orthogonal, self-dual and ACD codes over finite fields through additive GRS and GTRS codes. We also obtain several classes of additive TRS codes that are not monomially equivalent to additive RS codes. Based on additive MDS codes whose dual codes are also MDS, we provide a perfect threshold secret-sharing scheme that can detect cheating, identify a certain number of cheaters among the participants, and correctly recover the secret.