This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easily incorporated into fast algorithms that construct codes of desired girth between 6 and 12; our own algorithms are presented for each girth, together with constructions obtained from them and corresponding computer simulations. More importantly, this paper emphasizes how the girth conditions of the Tanner graph corresponding to a parity-check matrix composed of circulants relate to the matrix obtained by adding (over the integers) the circulant columns of the parity-check matrix. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square circulant submatrix of size 4 obtained from it.
翻译:4. 本文为在列重量为2和3的列中已经研究过的案例的毛细结果自然延伸提供了必要和充分的条件,而这种结果的依据是:以全成原样制成的准周期(QC)低密度对等检查(LDPC)码(LDPC),其基数分别为6、8、10和12,在具有列重量的对等检查矩阵中,其基数分别为6、8、10和12。这些结果是已经研究过的2和3列重量案例的毛细结果的自然延伸。 更重要的是,本文强调Tanner图的毛细数与由同级检查矩阵提供的对等检查矩阵之间的关系,通过添加(整数)该矩阵与产品在矩阵及其转换之间的权力特性。 gircurant条件可以很容易地纳入快速算法中,其中构建6至12之间期望的Girth 码;我们自己的算法是每个基数的,以及从它们得到的构造和相应的计算机模拟。更重要的是,本文强调Tanner图形的基数与由culant组成的对等数组成的矩阵的基数质矩阵的基数条件如何与通过添加(在整中)获得的基数表的基数的基数表的基数列的基数表的基数是分。