Generalized concatenated codes were introduced in the 1970s by Zinoviev. There are many types of codes in the literature that are known by other names that can be viewed as generalized concatenated codes. Examples include matrix-product codes, multilevel codes and generalized cascade codes. Decoding algorithms for generalized concatenated codes were developed during the 1970s and 1980s. However, their use does not appear to be as widespread as it should, especially for codes that are known by other names but can be viewed as generalized concatenated codes. In this paper we review the decoding algorithms for concatenated codes, generalized concatenated codes and matrix-product codes, and clarify the connection between matrix-product codes and generalized concatenated codes. We present a small improvement to the decoding algorithm for concatenated codes. We also extend the decoding algorithms from errors-only decoders to error-and-erasure decoders. Furthermore, we improve the upper bound on the computational complexity of the decoding algorithm in the case of matrix-product codes where the generator matrix for the inner code is non-singular by columns.

中文翻译：

---

关于广义级联码和矩阵乘积码的解码

季诺维也夫在 1970 年代引入了广义级联码。文献中有许多类型的代码以其他名称为人所知，可以将其视为广义级联代码。示例包括矩阵乘积码、多级码和广义级联码。广义级联码的解码算法是在 1970 年代和 1980 年代开发的。然而，它们的使用似乎并不像应有的那样广泛，特别是对于以其他名称已知但可以被视为广义级联代码的代码。在本文中，我们回顾了级联码、广义级联码和矩阵乘积码的解码算法，并阐明了矩阵乘积码和广义级联码之间的联系。我们对级联码的解码算法进行了小幅改进。我们还将解码算法从仅错误解码器扩展到错误和擦除解码器。此外，我们改进了在矩阵乘积码的情况下解码算法的计算复杂度的上限，其中内码的生成矩阵按列是非奇异的。