On the distances of cyclic codes of length \(2^e\) over \(\mathbb Z_4\) (Q1045137)

Cyclic codes of length \(2^e\) over \(\mathbb{Z}_4\) are precisely the ideals of the local ring \(\mathbb{Z}_4[X]/(X^{2^e}-1)\); their structure was determined by \textit{T. Abualrub} and \textit{R. Oehmke} (see for instance [Discrete Appl. Math. 128, No. 1, 3--9 (2003; Zbl 1025.94022)] or the article by \textit{S. Dougherty} and \textit{S. Ling} [Des. Codes Cryptography 39, No. 2, 127--153 (2006; Zbl 1172.94637)]). Using these results, the authors investigate the Hamming and the Lee distances of such codes; they give a complete list of the distances with exact results in nearly all 23 cases; exceptions are the Lee distances of three cases for which upper and lower bounds are given.