On Existence of Good Self-Dual Quasi-Cyclic Codes
IEEE Transactions on Information Theory
For a long time, asymptotically good self-dual codes have been known to exist. Asymptotically good 2-quasi-cyclic codes of rate 1 2 have also been known to exist for a long time. Recently, it was proved that there are binary self-dual 3-quasi-cyclic codes of length asymptotically meeting the Gilbert-Varshamov bound. Unlike 2-quasi-cyclic codes, which are defined to have a cyclic group of order 2 as a subgroup of their permutation group, the 3-quasi-cyclic codes are defined with a permutation
... up of fixed order of 3. So, from the decoding point of view, 2-quasi-cyclic codes are preferable to 3-quasi-cyclic codes. In this correspondence, with the assumption that there are infinite primes with respect to (w r t.) which 2 is primitive, we prove that there exist classes of self-dual 2 -quasi-cyclic codes and Type II 8 -quasi-cyclic codes of length respectively 2 and 8 which asymptotically meet the Gilbert-Varshamov bound. When compared with the order of the defining permutation groups, these classes of codes lie between the 2-quasi-cyclic codes and the 3-quasi-cyclic codes of length , considered in previous works. Abstract-Using representation theoretical methods we investigate self-dual group codes and their extensions in characteristic 2. We prove that the existence of a self-dual extended group code heavily depends on a particular structure of the group algebra which can be checked by an easy-to-handle criteria in elementary number theory. Surprisingly, in the binary case such a code is doubly even if the converse of Gleason's theorem holds true, i.e., the length of the code is divisible by 8. Furthermore, we give a short representation theoretical proof of an earlier result of Sloane and Thompson which states that a binary self-dual group code is never doubly even if the Sylow 2-subgroups of are cyclic. It turns out that exactly in the case of a cyclic or Klein four group as Sylow 2-subgroup doubly even group codes do not exist.