On the lifted Melas code

Abstract

The binary Melas code is a cyclic code with generator polynomial g(u)=p(u)p(u)^∗ where p(u) is a primitive polynomial of odd degree m≥5 and the ∗ denotes reciprocation. The even-weight subcode of a Melas code has generator polynomial (u+1)g(u) and parameters [2^m−1,2^m−2m−2,6]. This code is lifted to $\mathbb {Z}_{4}$ and the quaternary code is shown to have parameters [2^m−1,2^m−2m−2,d _L ≥8], where d _L denotes the minimum Lee distance. An algebraic decoding algorithm correcting all errors of Lee weight ≤3 is presented for this code. The Gray map of this quaternary code is a binary code with parameters [2^m+1−2,2^m+1−4m−4,d _H ≥8] where d _H is the minimum Hamming distance. For m=5,7 the minimum distance equals the minimum distance of the best known linear code for the given length and code size.