An explicit expression for Euclidean self-dual cyclic codes over F2m+uF2m of length 2s

Abstract Let F 2 m be the finite field of 2 m elements and s be any positive integer. The existing literature only gives an effective calculation method to represent all distinct Euclidean self-dual cyclic codes of length 2 s over the finite chain ring F 2 m + u F 2 m ( u 2 = 0 ) , such as in Cao et al., (2019). As a development of this topic, we provide an explicit expression for each of these self-dual cyclic codes, using binomial coefficients. The Gray image of any self-dual cyclic code over F 2 m + u F 2 m of length 2 s is a self-dual 2 -quasi-cyclic code over F 2 m of length 2 s + 1 . In particular, we give a generator matrix for each of these self-dual 2 -quasi-cyclic codes over F 2 m .