On Binary Linear Codes Whose Automorphism Group is Trivial

Abstract

We describe an important family of linear codes with trivial automorphism group. Two new concepts are defined: the smallest code of trivial automorphism group (smallest TAG-code) and the notion of minimal suhcode with trivial automorphism group (minimal TAG-subcode). We show in this work that the smallest TAG-code is unique up to equivalence and that there exist codes without a minimal TAG-suhcodes. We give an example of two minimal TAG-suhcodes of the same code that are not equivalent. We also construct for any lengthn≥6 a [2n, n, 3]2 code with trivial automorphism group. Finally we prove the nonexistence of constant weight codes (CW-code) with trivial automorphism group. Some theoritical software tools are used.