On the lifted Zetterberg code

Abstract

The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial $$g(x)=(x+1)p(x)$$g(x)=(x+1)p(x), where p(x) is the minimum polynomial over GF(2) of an element of order $$2^m+1$$2m+1 in $$GF(2^{2m})$$GF(22m) and m is even. This even binary code has parameters $$[2^m+1,2^m-2m, 6]$$[2m+1,2m-2m,6]. The quaternary code obtained by lifting the code to the alphabet $${\mathbb {Z}}_4=\{0,1,2,3\}$$Z4={0,1,2,3} is shown to have parameters $$[2^m+1,2^m-2m, d_L ]$$[2m+1,2m-2m,dL], where $$d_L \ge 8$$dL?8 denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters $$(2^{m+1}+2,2^k,d_H)$$(2m+1+2,2k,dH), where $$d_H \ge 8$$dH?8 denotes the minimum Hamming weight and $$k=2^{m+1}-4m$$k=2m+1-4m. For $$m=6$$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 $${\mathbb {Z}}_4$$Z4-codes for all even m. This appears to be the first infinite family of $${\mathbb {Z}}_4$$Z4-codes of length $$n=2^m+1$$n=2m+1 with $$d_L \ge 8$$dL?8 having an algebraic decoding algorithm.