Additive cyclic codes over $\mathbb F_4$

Abstract

In this paper we find a canonical form decomposition for additive cyclic codes of odd length over $\mathbb F_4$. This decomposition is used to count the number of such codes. We also reprove that each code is the $\mathbb F_2$-span of at most two codewords and their cyclic shifts, a fact first proved in [2]. A count is given for the number of codes that are the $\mathbb F_2$-span of one codeword and its cyclic shifts. We can examine this decomposition to see precisely when the code is self-orthogonal or self-dual under the trace inner product. Using this, a count is presented for the number of self-orthogonal and self-dual additive cyclic codes of odd length. We also provide a count of the additive cyclic and additive cyclic self-orthogonal codes as a function of their $\mathbb F_2$-dimension.