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Erschienen in:
01.01.2016
On the lifted Melas code
Erschienen in: Cryptography and Communications | Ausgabe 1/2016
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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 [2
m
−1,2
m
−2m−2,6]. This code is lifted to \(\mathbb {Z}_{4}\) and the quaternary code is shown to have parameters [2
m
−1,2
m
−2m−2,d
L
≥8], where d
L
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 [2
m+1−2,2
m+1−4m−4,d
H
≥8] where d
H
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.