The Truncated, Upper-Triangle Pascal Matrix, Linear Independence and Results in Matroid Theory.

Let $q$ be a power of a prime $p$. Matrices over $F_q$ in which every subset of basis size of the columns are independent, are of interest in coding theory, matroid theory, and projective geometry. For any positive integer ${m \leq p}$ and bijection ${\sigma: N_{\leq q-1}\cup\{0\}\rightarrow F_q}$, we show that the $m \times (q+1)$ matrix $H_{q,m}$, with $\{U_{q}\}_{i,j} = {\sigma(j-1)\choose {i-1}}$, if $j \leq s$, $1$ if $j = s + 1$ and $i = m$ and $0$ if otherwise is such a matrix. We term such matrices "supplemented pascal matrices" and demonstrate that they are Reed-Solomon codes. Further, we describe applications in matroid theory.