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Published in:
26-11-2021 | Original Research
Linear codes with eight weights over \(\mathbb {F}_p+u\mathbb {F}_p\)
Published in: Journal of Applied Mathematics and Computing | Issue 5/2022
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Abstract
In this article, based on the defining set \(D=\{x\in \mathbb {F}_{p^m}^*:Tr(x)=0\}\), we explore the Lee-weight distribution of linear codes \({\mathcal {C}}_D=\{(tr(ax^2))_{x\in D}:a\in \mathbb {F}_{p^m}+u\mathbb {F}_{p^m}\}\) over the finite ring \(\mathbb {F}_p+u\mathbb {F}_p\) with p being an odd prime and \(u^2=0\). By employing the exponential and Gauss sums, we calculate the Lee weight of all possible codewords as well as their frequencies. Two classes of eight-weight linear codes are obtained, where one of them is new. We also show that for some small values m, the code \({\mathcal {C}}_D\) has two weights (\(m=2\)) and seven weights (\(m=3,4\) and \(p=3\)).