Syndrome is Zero iff Vector is Codeword/Mistake

Source Work

1996: John F. Humphreys: A Course in Group Theory: Chapter $6$: Error-correcting codes

Proposition $6.20$: Proof

Mistake

Hence, the element $c$ is a codeword if and only if $c = u G = u \paren {\begin{array} {c|c} I & A \end{array} } = \paren {\begin{array} {c|c} u & v \end{array} }$, where $v = u A$, and $\paren {\begin{array} {c|c} U & v \end{array} }$ denotes the concatenation of $u$ with $v$, ...

Correction

Where it says:

where ... $\paren {\begin{array} {c|c} U & v \end{array} }$ denotes the concatenation of $u$ with $v$

it should have:

$\paren {\begin{array} {c|c} u & v \end{array} }$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Proposition $6.20$