Constructions of Polynomial Codes Based on Circular Permutation Over Finite Fields

Xiuli Wang,Yakun Hao,Dan Qiao

doi:10.1109/access.2020.3010534

Xiuli Wang, Yakun Hao + Show 1 more

Open Access

https://doi.org/10.1109/access.2020.3010534

Copy DOI

Abstract

As a kind of very important error-correcting codes, polynomial codes have the advantages of good performance and simple structure. Generalized R-S codes over F q as polynomial codes have the advantages described above, but their lengths n are at most q. In order to increase the length of the codes, in this paper, we apply a method similar to the construction of the generalized R-S codes over F q and develop Ling and Xing's idea to construct new q-ary linear codes. First, based on the idea of constructing polynomials codes over F q , we construct a class of polynomial functions by using the circular permutation defined over F q , and a new type of linear codes is constructed from the properties of these functions and Mobius functions. Then, we give the value of the length n, the dimension k and the lower bound of the minimum distance d of the new linear codes. It turns out that some codes from this construction have good parameters based on Brouwer's table.

Highlights

Since Shannon started the error-correcting codes theory in 1948, the research of error-correcting codes theory has been developed for more than 60 years [1]
Polynomial codes are an important class of error-correcting codes, they have good error-correcting ability and simple structure
As polynomial codes, generalized Reed-Solomon codes [2] over the finite field Fq, have the following construction: Let q be a prime power, n, k be two integers with 1 < n ≤ q and 1 < k ≤ n, and let Pk be the set of all polynomials in Fq[x] with degree less than k

Summary

Introduction

Since Shannon started the error-correcting codes theory in 1948, the research of error-correcting codes theory has been developed for more than 60 years [1]. Xing constructed codes by evaluating the symmetric polynomials over Fq at the elements in the extension field Fqs of Fq, where s is an arbitrary positive integer [5].

Results

Conclusion