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05-12-2019 | Original Paper | Issue 4/2021

Applicable Algebra in Engineering, Communication and Computing 4/2021

On self-duality and hulls of cyclic codes over \(\frac{\mathbb {F}_{2^m}[u]}{\langle u^k\rangle }\) with oddly even length

Journal:
Applicable Algebra in Engineering, Communication and Computing > Issue 4/2021
Authors:
Yonglin Cao, Yuan Cao, Fang-Wei Fu
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Abstract

Let \(\mathbb {F}_{2^m}\) be a finite field of \(2^m\) elements and denote \(R=\mathbb {F}_{2^m}[u]/\langle u^k\rangle \) \(=\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+\cdots +u^{k-1}\mathbb {F}_{2^m}\) (\(u^k=0\)), where k is an integer satisfying \(k\ge 2\). For any odd positive integer n, an explicit representation for every self-dual cyclic code over R of length 2n and a mass formula to count the number of these codes are given. In particular, a generator matrix is provided for the self-dual 2-quasi-cyclic code of length 4n over \(\mathbb {F}_{2^m}\) derived by an arbitrary self-dual cyclic code of length 2n over \(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\) and a Gray map from \(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\) onto \(\mathbb {F}_{2^m}^2\). Finally, the hull of each cyclic code with length 2n over \(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\) is determined and all distinct self-orthogonal cyclic codes of length 2n over \(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\) are listed.

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