Compute $s_1 = [I_{12} | B]r^T$. If $s_1 = 0$, then accept $r$; STOP.

If $1 \leq w(s_1) \leq 3$, then set $e = (s_1^T, 0)$ and decode $r$ to $c = r - e$ and STOP.

Compare $s_1$ to the rows of $B$. If row $i$ of $B$ differs in exactly one position (say $j$) or two positions (say $j$ and $k$), then correct $x$ in position $j$, or positions $j$ and $k$; correct $y$ in position $i$; STOP

Compute $s_2 = [B | I_{12}]r^T$. If $w(s_2) \leq 3$, then set $e = (0, s_2^T)$ and decode $r$ to $c = r - e$ and STOP.

Compare $s_2$ to the rows of $B$. If row $i$ of $B$ differs in exactly one position (say $j$) or two positions (say $j$ and $k$), then correct $y$ in position $j$, or positions $j$ and $k$; correct $x$ in position $i$; STOP

Reject $r$