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 [2m?1,2m?2m?2,6]. This code is lifted to ?4$\mathbb {Z}_{4}$ and the quaternary code is shown to have parameters [2m?1,2m?2m?2,dL?8], where dL 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 [2m+1?2,2m+1?4m?4,dH?8] where dH 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.