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Erschienen in:
17.01.2021 | Original Paper
A new class of distance-optimal binary cyclic codes and their duals
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 1/2023
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Abstract
Let \(m=8k\) and \(\alpha\) be a primitive element of the finite field \({{\mathbb {G}}{\mathbb {F}}}(2^m)\), where \(k\ge 2\) is an integer. In this paper, a class of binary cyclic codes \({{\mathcal {C}}}_{(u,v)}\) of length \(2^m-1\) with two nonzeros \(\alpha ^{-u}\) and \(\alpha ^{-v}\) is studied, where \((u,v)=(1,(2^{m}-1)/17)\). It turns out that \({{\mathcal {C}}}_{(u,v)}\) has parameters \([2^m-1,2^m-m-9,4]\) and is distance-optimal with respect to the Sphere Packing bound. The weight distribution of the dual of \({{\mathcal {C}}}_{(u,v)}\) is also completely determined based on some results on Gaussian periods.