This article presents the construction of binary linear Hadamard codes with parameters (2n,4n,n) over the fields F₄ and F₈. We generalize this construction to the field \(F_{2^{k}}\) to generate Hadamard codes with parameters \(\left (2^{k-1}n, 2^{k}n, 2^{k-2}n\right )\) for \(k\in \mathbb {N}\). We construct s-PD sets of size s + 1, a special subset of the permutation automorphism group of a code, which enables the correction of s errors for Hadamard codes over the field F₄. We also discuss the decoding algorithm for these Hadamard codes.

中文翻译：

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二进制Hadamard码及其s -PD集的构造

本文介绍了在字段F ₄和F _8上具有参数（2 n，4 n，n）的二进制线性Hadamard码的构造。我们将此构造推广到字段\（F_ {2 ^ {k}} \）以生成具有参数\（\ left（2 ^ {k-1} n，2 ^ {k} n，2 ^ {k -2} n \ right）\）表示\（k \ in \ mathbb {N} \）。我们构建小号-PD集大小的小号+ 1，置换构群的码，这使得校正的一个特殊的子集小号为Hadamard码错误在字段˚F ₄。我们还将讨论这些Hadamard码的解码算法。