Consider the binary Hamming code $\text{Ham} (\mathbb{Z}_2^5)$ and suppose that under this encoding we send the codeword $x$ down a symmetric channel with symbol error probability $p$.
Using nearest neighbour decoding, what is the probability (in terms of $p$) that a sent codeword is decoded incorrectly?
Using syndrome decoding (which I believe is a form of nearest neighbor decoding), we find that a codeword $x$ is decoded correctly iff $S(x) = S(y)$ where $y$ is coset leader of the code used.
For $\text{Ham} (\mathbb{Z}_2^5)$ the coset leaders are exactly the vectors $\mathbb{Z}_2^5$ of weight less than or equal to 1. We may denote these coset leaders by $$ e_i = 00 \dots 010 \dots 00 $$ where the $i$ is in the $i^\text{th}$ position (for $i = 1,2, \dots, 5$). Then, assuming that the columns the Hamming matrix for $\text{Ham} (\mathbb{Z}_2^5)$ are the vectors in $\mathbb{Z}_2^5 \backslash \{ \underline{0} \}$ ordered in lex order, the syndrome of each coset leader $e_i$ is given by $$ S(e_i) = e_i H^T = (0, \dots, 0, 1, 0, \dots, 0) \left( \begin{matrix} c_1 \\ \vdots \\ c_i \\ \vdots \\ c_n \end{matrix} \right) = c_i $$ where $c_i$ denotes the $i^\text{th}$ column of the Hamming matrix (which are listed as the rows of the transpose matrix $H^T$). Further, since we have ordered the columns of the Hamming matrix lexographically, we find that $c_i$ is just the integer $i$ expressed in binary (with 5 digits).
Thus, the probability that a $x \in \text{Ham} (\mathbb{Z}_2^5)$ is decoded correctly is equal to the probability that $S(x) = c_i$ for some $i$.
I'm not sure if I'm going the right way in solving this, but if so I am not sure where to go from here.