On the lifted Zetterberg code (original) (raw)

2015, Designs, Codes and Cryptography

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 G F(2) of an element of order 2 m + 1 in G F(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 Z 4 = {0, 1, 2, 3} is shown to have parameters [2 m + 1, 2 m − 2m, d L ], where d L ≥ 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 ≥ 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 Z 4-codes for all even m. This appears to be the first infinite family of Z 4-codes of length n = 2 m + 1 with d L ≥ 8 having an algebraic decoding algorithm. Keywords Zetterberg code • Cyclic codes • Codes over Z 4 Mathematics Subject Classification 94B15 • 94B35 1 Introduction Codes over Z 4 , the ring of integers modulo 4, have been intensively studied during the last 25 years using the theory of Galois rings, in particular after the ideas developed in the 1990s Communicated by J. Bierbrauer.