The matrix representations of linear codes have been well-studied for use as disjunct matrices. However, no connection has previously been made between the properties of disjunct matrices and the parity-check codes obtained from them. This paper makes this connection for the first time. We provide some fundamental results on parity-check codes from general disjunct matrices (in particular, a minimum distance bound). We then consider three specific constructions of disjunct matrices and provide parameters of their corresponding parity-check codes including rate, distance, girth, and density. We show that, by choosing the correct parameters, the codes we construct have the best possible error-correction performance after one round of bit-flipping decoding with regard to a modified version of Gallager's bit-flipping decoding algorithm.

翻译：线性码的矩阵表示已被深入研究并用作分离矩阵。然而，此前尚未将分离矩阵的性质与由此类矩阵得到的奇偶校验码建立联系。本文首次建立了这一联系。我们给出了从一般分离矩阵构造奇偶校验码的一些基础性结论（尤其是最小距离界）。随后，我们研究了三种具体的分离矩阵构造方法，并给出了其对应奇偶校验码的参数，包括码率、距离、围长和密度。研究表明，通过选择合适的参数，我们构造的码在经改进的Gallager比特翻转译码算法一轮迭代后，可获得最优的纠错性能。