The impact of alternant codes over Eisenstein integers on modern technology

Abstract

In this article, we are going to discuss about how alternant codes over Eisenstein integers are crucial in today’s technologies especially in encoding data and correction of those data. Based on the parity check matrices of alternant codes, the codes are constructed with symbols from finite commutative local rings with unity of Eisenstein integers modulo $n$. Particularly, the factorization of $x^n-1$ is over the unit-ring of an appropriate extension of the finite local commutative ring of Eisenstein integers. These codes are built similarly to the BCH codes over the finite integer rings and provide a solid background for the process of encoding the data. This work further proved the applicability of the said algorithm for the purpose of error correction in the discussed alternant codes in order to improve transference effectiveness. Through analyzing the mathematical concepts behind as well as practical applications of alternant codes over Eisenstein integers the possibility of changing today’s technological setting is disclosed. The paper demonstrates that the purpose of the codes is to enhance data integrity and security and underlines that this is why they should be useful for modern communication systems and much more.

Publication
Computational and Applied Mathematics, 44, 95. doi:10.1007/s40314-024-03057-y

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