On the lifted Zetterberg code - Alahmadi, Adel; Alhazmi, Hussain; Helleseth, Tor; Hijazi, Rola; Muthana, Najat; Solé, Patrick (Designs, codes, and cryptography, 2016)
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 G F ( 2 2 m ) and m is even. This even binary code has parameters [ 2 m + 1 , 2 m - 2 m , 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 - 2 m , 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 - 4 m . 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.