Memory maximum-distance-separable (mMDS) sliding window codes are a type of erasure codes with high erasure-correction capability and low decoding delay. In this paper, we study two types of systematic mMDS sliding window codes over erasure channels, i.e., scalar codes defined over a finite field $GF(2^{L})$ , and vector codes defined over a vector space $GF(2)^{L}$ . We first devise an efficient heuristic algorithm to produce an mMDS sliding window scalar code over relatively small $GF(2^{L})$ . Then, we investigate a special class of mMDS sliding window vector codes whose encoding/decoding are achieved by basic circular-shift and bit-wise XOR operations, and propose a general method to generate such mMDS vector codes. Our complexity analysis shows that the proposed vector codes yield much lower encoding/decoding complexity than the scalar codes. The theoretical and numerical results also demonstrate that mMDS sliding window codes dominate MDS block codes in terms of decoding delay and erasure-correction capability.

中文翻译：

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擦除通道上的系统内存MDS滑动窗口代码

存储器最大距离可分离（mMDS）滑动窗口代码是一种擦除代码，具有较高的擦除校正能力和较低的解码延迟。在本文中，我们研究了擦除通道上的两种类型的系统mMDS滑动窗口代码，IE， 在有限域上定义的标量代码 $ GF（2 ^ {L}）$ ，以及在向量空间上定义的向量代码 $ GF（2）^ {L} $ 。我们首先设计一种有效的启发式算法，以在相对较小的范围内生成mMDS滑动窗口标量代码 $ GF（2 ^ {L}）$ 。然后，我们研究了一类特殊的mMDS滑动窗口矢量代码，其编码/解码是通过基本的循环移位和按位XOR运算实现的，并提出了一种生成此类mMDS矢量代码的通用方法。我们的复杂度分析表明，与标量代码相比，提出的矢量代码产生的编码/解码复杂度要低得多。理论和数值结果还表明，就解码延迟和擦除校正能力而言，mMDS滑动窗口代码在MDS块代码中占主导地位。