On the idempotents of cyclic codes over $\mathbb{F}_{2^t}$
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초록
We study cyclic codes of length $n$ over $\mathbb{F}_{2^t}$. Cyclic codes can be viewed as ideals in $\mathcal{R}_n = \mathbb{F}_{2^t}[x]/(x^n - 1)$. It is known that there is a unique generating idempotent for each ideal. Let $e(x) \in \mathcal{R}_n$. If $t = 1$ or $t = 2$, then there is a necessary and sufficient condition that $e(x)$ is an idempotent. But there is no known similar result for $t \geq 3$. In this paper we give an answer for this problem.
Abstract
We study cyclic codes of length $n$ over $\mathbb{F}_{2^t}$. Cyclic codes can be viewed as ideals in $\mathcal{R}_n = \mathbb{F}_{2^t}[x]/(x^n - 1)$. It is known that there is a unique generating idempotent for each ideal. Let $e(x) \in \mathcal{R}_n$. If $t = 1$ or $t = 2$, then there is a necessary and sufficient condition that $e(x)$ is an idempotent. But there is no known similar result for $t \geq 3$. In this paper we give an answer for this problem.
참고문헌 (3)
- W. C. Huffman, "Fundamentals of Error-correcting Codes" (2003), Cambridge University Press
- W. W. Peterson, "Error-Correcting Codes" (1961), MIT Press
- W. W. Peterson, "Error-Correcting Codes" (1972), MIT Press