On the hull and complementarity of certain quasi-cyclic codes

Official URL: https://risc01.sabanciuniv.edu/record=b2771062_(Table of contents)

Linear codes with small hull dimension have been of interest due to their applications to various problems in coding theory and cryptography. Linear complementary dual codes, which are codes with zero hull dimension, and their generalization called linear complementary pair of codes have also been studied widely in the literature. We study these notions for quasi-cyclic codes. We show that all admissible hull dimensions for quasi-cyclic codes, according to their CRT decomposition, are attained. We pay particular attention to double and four circulant codes, which are one and two generator quasi-cyclic codes of special form. We formulate the hull dimension for these families in terms of the polynomials involved in their generating elements. We obtain results on possible hull dimensions, such as the hull of a four circulant code being even and the nonexistence of hull dimension one double circulant codes over Fq if q ≡ 3 (mod 4). We present numerical results on the parameters of double and four circulant codes with extra conditions, such as having fixed small hull dimension or being complementary dual. We also enumerate double and four circulant codes with zero or the smallest possible positive hull dimension and prove that double circulant codes with zero or one hull dimension are asymptotically good.