Decoding $p$-ary Hamming codes

Here's the decoding algorithm: Let $y$ be the received vector. Let

$${\bf e}_1=(1,0,...,0), \ \ {\bf e}_2=(0,1,0,...,0), ..., {\bf e}_n=(0,...,0,1),$$

be the standard basis vectors in $\mathbb{F}_p^n$. Assume that $y$ has $\leq 1$ error.

(1) Compute $y\cdot H^t$. This is an $n$-tuple, so it must be of the form $c\cdot {\bf s}$, for some ${\bf s}\in S$ and some $c\in \mathbb{F}_p^\times$.

(2) If ${\bf s}$ is the $i^{th}$ element of $S$ then the decoded vector is ${\bf y}-c\cdot {\bf e}_i$.

Exercise 3.9.9   Decode $(0,1,0,0,0,0,1)$ in the binary Hamming code of §3.4.2 using this algorithm.

Exercise 3.9.10   Decode $(0,0,1,1)$ in the $3$-ary $(4,2,3)$-Hamming code of Example 3.9.5 using this algorithm.