Advances in Mathematics of Communications (AMC)

Additive cyclic codes over $\mathbb F_4$

Pages: 427 - 459, Volume 1, Issue 4, November 2007      doi:10.3934/amc.2007.1.427

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W. Cary Huffman - Department of Mathematics and Statistics, Loyola University, Chicago, IL 60626, United States (email)

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.

Keywords:  Cyclic codes, additive codes, self-orthogonal codes, self-dual codes.
Mathematics Subject Classification:  Primary: 94B15.

Received: May 2007;      Revised: October 2007;      Available Online: October 2007.