The Decoding Error Probability of Linear Codes Over the Erasure Channel

Generalized Column Distances

“…Since its introduction in the context of block codes, the generalized Hamming weights are perceived as a refinement of the minimal distance. This perception was recently re-enforced by the work [34] where it is shown that the error probability of a linear code over the erasure channel under list decoding or maximum likelihood decoding is expressed by its support weight distributions. We think that a challenging work would be to understand these results considering convolutional codes.…”

“…A more precise role of the generalized weight hierarchy has been recently investigated to derive error probabilities after decoding of block codes over the erasure channel. This was first approached by Didier [12] and later developed in [11], [33] and finally in [34], were an expression of the unsuccessful decoding probability under list decoding and unambiguous decoding was established in terms of the distribution of generalized weights 1 .…”

Decoding error probability in the erasure channel and the $\boldsymbol~r$-th support weight distribution

“…(参见文献 [2][3][4]) 中, 因而线性码在擦除信道中的译码问题受到很多学者的关注. 在最近几年, 擦除信 道的线性码的三大重要译码原则, 即明确译码、最大似然译码和列表译码, 已经被提出并得到发展; 相 应的译码错误概率问题, 即按照译码原则对于从擦除信道中传来的信息进行错误译码的概率问题, 已 经被众多学者研究, 相应的引人注目的结果可参见文献 [5][6][7][8].…”

“…特别地, 2019 年, Shen 和 Fu [7] 运用他们新近提出的线性码的 "q ℓ -不可修正集" 的概念并且改进 文献 [6,8] 中的研究结果, 详尽地研究了擦除信道中的线性码的译码错误概率问题. 在众多解决的问 题以外, 他们得到了线性码在擦除信道中基于以上三大译码原则下的译码错误概率的显式表达式.…”

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Inspired by a recent work of Shen and Fu (2019), we study the decoding error probability of linear codes over the erasure channel under three decoding principles. We first simplify the explicit formulas obtained by Shen and Fu (2019), showing a much more direct relation to the r-th support weight distribution of linear codes. Then we compute the average r-th support weight distribution of [n, k]q linear codes.

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