基于有限域结构的LRC码的存在性讨论

Discussion on Existence of LRC Codes Based on Finite Field Structure

假设C是有限域Fq上的(n,k)线性码,若码字的每个分量值是其他r个分量值的函数,则称C为(n,k,r)LRC码,这里r相对于码长来说是个较小的数。基于有限域结构构造LRC码的方法通常有3种:利用有限域的加法结构、乘法结构及其子域上的向量空间结构。然而,这些构造方法不是对任意局部参数为r的LRC码都能构造。为了解决这个问题,本文通过组合代数等方法,对任意给定素数p,提出了Fq上G-多项式存在的充分条件,讨论了一类局部参数r=p2+p-1的局部恢复码的存在条件,并通过两个实例来说明相关问题。

Supposing C to be a(n,k)linear code over a finite field Fq,it is called the(n,k,r)LRC code if each compo-nent value of the codeword is a function of other r component values,where r is a smaller number compared to the length of the code.There are usually three ways to construct LRC codes based on finite field structure,which can be listed as:using the addition structure,multiplicative construction of the finite field and vector space construction of its subfield.However,these construction methods are not suitable for any expected LRC code with local parameter r.In order to solve this problem,this ar-ticle uses methods including algebraic combinatorics to propose the sufficient condition of G-polynomial’s existence on Fq in terms of any given prime number p.The existing prerequisite of the local recovery code for a class of local parameter r=p2+p-1 is further discussed.Finally,two instances are used to illustrate the relevant problem.