Lee Distance of cyclic codes of length $2^\varsigma$ over $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$

Divya Acharya; Prasanna Poojary; Vadiraja Bhatta G R

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arXiv · 2509.03911

Lee Distance of cyclic codes of length $2^\varsigma$ over $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$

Divya Acharya·Prasanna Poojary·Vadiraja Bhatta G R

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Abstract

Let $p$ be a prime number and $\varsigma$ and $m$ be a positive integers. Let $\mathcal{R} = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$ ($u^3 = 0$). Cyclic codes of length $2^\varsigma$ over $\mathcal{R}$ are precisely the ideals of the local ring $\frac{\mathcal{R}[x]}{\langle x^{2^\varsigma}-1 \rangle}$. The Gray map from a code of Lee weight over $\mathbb{Z}_4$ to a code with Hamming weight over $\mathbb{F}_2$ is known to preserve weight. In this paper, we determine the Lee distance of cyclic codes of length $2^\varsigma$ over $\mathcal{R}$.

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Divya Acharya, Prasanna Poojary, Vadiraja Bhatta G R. 2025-09-04. Lee Distance of cyclic codes of length $2^\varsigma$ over $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$. https://arxiv.org/abs/2509.03911

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