论文标题
提起的多重代码
Lifted Multiplicity Codes
论文作者
论文摘要
提起的芦苇 - 固体代码和多重性代码是两类评估代码，允许设计高速代码，可以从许多脱节集中恢复每个代码字或信息符号。最近，已经将基本方法组合在一起以构建提起的双变量多样性代码，从而可以进一步提高速率。我们通过在任意数量的变量中从多项式获得的提升的多重性代码的速率和距离提供下限来继续研究这些代码。具体而言，我们调查了由$ m $变量单元的线性跨度形成的提起的多重代码的子代码，该跨度限制在$ \ mathbb {f} _q^m $中的任意线的限制等于低程度的UNI-Variate polynomial。我们发现，当变量$ m $固定并且字母尺寸$ q = 2^\ ell $很大时，此类单元的分数的紧密渐近行为是大的。对于某些参数制度，然后证明抬高的多重代码在冗余和每个代码字的脱节恢复集之间的折衷程度更好，而不是以前已知的构造。此外，我们提出了一种局部自我纠正算法，用于升起的多重性代码。
Lifted Reed-Solomon codes and multiplicity codes are two classes of evaluation codes that allow for the design of high-rate codes that can recover every codeword or information symbol from many disjoint sets. Recently, the underlying approaches have been combined to construct lifted bi-variate multiplicity codes, that can further improve on the rate. We continue the study of these codes by providing lower bounds on the rate and distance for lifted multiplicity codes obtained from polynomials in an arbitrary number of variables. Specifically, we investigate a subcode of a lifted multiplicity code formed by the linear span of $m$-variate monomials whose restriction to an arbitrary line in $\mathbb{F}_q^m$ is equivalent to a low-degree uni-variate polynomial. We find the tight asymptotic behavior of the fraction of such monomials when the number of variables $m$ is fixed and the alphabet size $q=2^\ell$ is large. For some parameter regimes, lifted multiplicity codes are then shown to have a better trade-off between redundancy and the number of disjoint recovering sets for every codeword or information symbol than previously known constructions. Additionally, we present a local self-correction algorithm for lifted multiplicity codes.
