Suppose that $m,k,t$ are integers with $m,k\ge 1$ and $0\le t$ and A is the $k\times k$ matrix with the $(i,j)$-entry $(\genfrac{}{}{0ex}{}{t+(j-1)m}{i-1})$. When t = 0 and m = 1, this is the upper triangular Pascal matrix. Here, first we study the properties of this matrix, in particular, we find its determinant and its LDU decomposition and also study its inverse. Then by using this matrix we present a generalization of the Mattson-Solomon transform and a polynomial formulation for its inverse to the case that the sequence length and the characteristic of the base field are not coprime. At the end, we use this generalized Mattson-Solomon transform to present a lower bound on the length of repeated root cyclic codes, which can be seen as a generalization of the BCH bound.

假设$米,k,吨$是整数$米,k\ge \mathrm{1个}$和$0\le 吨$A是_$k\times k$矩阵与$(我,j)$-入口$(\genfrac{}{}{0ex}{}{吨+(j-\mathrm{1个})米}{我-\mathrm{1个}})$. 当t  = 0 且m  = 1 时，这是上三角帕斯卡矩阵。在这里，首先我们研究这个矩阵的性质，特别是，我们找到它的行列式和它的 LDU 分解，并研究它的逆矩阵。然后通过使用该矩阵，我们提出了 Mattson-Solomon 变换的推广及其逆向序列长度和基域的特征不互质的情况的多项式公式。最后，我们使用这个广义的 Mattson-Solomon 变换来给出重复根循环码长度的下界，这可以看作是 BCH 界的推广。