Concatenated Structure and Construction of Certain Code Families

Elif Saçıkara

PhD Dissertation, 2018

Thesis Jury

Prof. Dr. Cem Güneri, (Thesis Advisor),

Prof. Dr. Erkay Savaş ,

Assoc. Prof. Kagan Kurşungöz,

Prof. Dr. Ferruh Özbudak,

Dr. Martino Borello

Date & Time: 25th, 2018– 3PM

Place:  FENS L029

Keywords : Concatenated codes, generalized concatenated codes, quasi-cyclic codes, generalized quasi-cyclic codes, quasi-abelian codes, linear complementary dual codes, linear complementary pair of codes

Abstract

In this thesis, we consider concatenated codes and their generalizations as the main tool for two different purposes. Our first aim is to extend the concatenated structure of quasi-cyclic codes to its two generalizations: generalized quasi-cyclic codes and quasi-abelian codes. Concatenated structure have consequences such as a general minimum distance bound. Hence, we obtain minimum distance bounds, which are analogous to Jensen's bound for quasi-cyclic codes, for generalized quasi-cyclic and quasi-abelian codes. We also prove that linear complementary dual

quasi-abelian codes are asymptotically good, using the concatenated structure. Moreover, for generalized quasi-cyclic and quasi-abelian codes, we prove, as in the quasi-cyclic codes, that their concatenated decomposition and the Chinese Remainder decomposition are equivalent.

The second purpose of the thesis is to construct a linear complementary pair of codes using concatenations. This class of codes have been of interest recently due to their applications in cryptography. This extends the recent result of Carlet et al. on the concatenated construction of linear complementary dual codes.