Mathematics Research Institute


Some Recent Results on LCD Codes

Buket Ozkaya (Nanyang Technological University, Singapore)

Fecha: 21/06/2019 13:00
Lugar: Seminario IMUVA. Edificio LUCIA

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. Several constructions of LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the rings were given in [3], along with a linear programming bound on the largest size of an LCD code of given length and minimum distance. In [4], the class of quasi-cyclic LCD codes was shown to be "good", by using their concatenated structure and a characterization of Hermitian LCD codes. Explicit constructions from codes over larger alphabets were also given. Analogous results are obtained for generalized quasicyclic codes in [5]. More recently, a class of one-generator quasi-twisted codes, namely LCD multinegacirculant codes, are studied [1]. Their concatenated structure yields exact enumeration results for index 2 and index 3, whereas for a general index t and co-index power of 2, a special enumeration is given which is needed for the asymptotic analysis by means of Dickson polynomials. In [2], analogous techniques were used to characterize and enumerate self-dual double negacirculant codes, which also have infinite families with relative distance satisfying a modified Gilbert-Varshamov bound. [1] A. Alahmadi, C. Güneri, B.Ö., H. Shoaib and P. Solé, "On linear complementary-dual multinegacirculant codes", to appear in Crypto. and Comm. [2] A. Alahmadi, C. Güneri, B.Ö., H. Shoaib and P. Solé, "On self-dual double negacirculant codes", Disc. Appl. Math., vol. 222, 205-212, 2017. [3] S.T. Dougherty, J.-L. Kim, B.Ö., L. Sok and P. Solé, "The combinatorics of LCD codes: Linear programming bound and orthogonal matrices", Int. J. Inf. and Cod. Theory, vol. 4, no. 2/3, 116-128, 2017. [4] C. Güneri, B.Ö., H. Shoaib and P. Solé, "Quasi-cyclic complementary dual codes", Finite Fields Appl., vol. 42, 67-80, 2016. [5] C. Güneri, F. Özbudak, B.Ö., E. Sacikara, Z. Sepasdar and P. Solé, "Structure and performance of generalized quasi-cyclic codes", Finite Fields and Their Applications, vol. 47, 183-202, 2017.