On the lifted Zetterberg code

Abstract

The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial $g(x)=(x+1)p(x)$, where p(x) is the minimum polynomial over GF(2) of an element of order $2^m+1$ in $GF(2^{2m})$ and m is even. This even binary code has parameters $[2^m+1,2^m-2m, 6]$. The quaternary code obtained by lifting the code to the alphabet ${\mathbb {Z}}_4=\{0,1,2,3\}$ is shown to have parameters $[2^m+1,2^m-2m, d_L ]$, where $d_L \ge 8$ denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters $(2^{m+1}+2,2^k,d_H)$, where $d_H \ge 8$ denotes the minimum Hamming weight and $k=2^{m+1}-4m$. For $m=6$ these parameters equal the parameters of the best known binary linear code for this length and dimension. Furthermore, a simple algebraic decoding algorithm is presented for these ${\mathbb {Z}}_4$-codes for all even m. This appears to be the first infinite family of ${\mathbb {Z}}_4$-codes of length $n=2^m+1$ with $d_L \ge 8$ having an algebraic decoding algorithm.