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@@ -148,15 +148,15 @@ The generator matrix for Reed-Solomon code is a Vandermonde matrix $V(a_1,a_2,\l
Fact: $V(a_1,a_2,\ldots,a_n)$ is invertible if and only if $a_1,a_2,\ldots,a_n$ are distinct. (that's how we choose $a_1,a_2,\ldots,a_n$)
The parity check matrix for Reed-Solomon code is also a Vandermonde matrix $V(a_1,a_2,\ldots,a_n)^T$ with scalar multiples of the columns.
The parity check matrix for Reed-Solomon code is also a Vandermonde matrix $V(a_1,a_2,\ldots,a_n)^\top$ with scalar multiples of the columns.
Some technical lemmas:
Let $G$ and $H$ be the generator and parity-check matrices of (any) linear code
$C = [n, k, d]_{\mathbb{F}_q}$. Then:
I. Then $H G^T = 0$.
II. Any matrix $M \in \mathbb{F}_q^{n-k \times k}$ such that $\rank(M) = n - k$ and $M G^T = 0$ is a parity-check matrix for $C$ (i.e. $C = \ker M$).
I. Then $H G^\top = 0$.
II. Any matrix $M \in \mathbb{F}_q^{n-k \times k}$ such that $\rank(M) = n - k$ and $M G^\top = 0$ is a parity-check matrix for $C$ (i.e. $C = \ker M$).
## Reed-Muller code