Covering Radius of linear code

Question

Covering Radius of a linear code: Show that if C is a linear [n, k, d] code over a finite field $F_{q}$, then R ≤ n − k.

Answer 1

"Hint":

Without loss of generality we can assume that the code is generated by a systematic generator matrix. This means that to any sequence of $k$ first symbols there exists a codeword beginning with that sequence. Upon reviewing the definition of the covering radius your task is to, given any vector, show that you can turn it into a codeword by changing at most $n-k$ components.

Answer 2

@Jyrki There you go.....

$G= [I_{k}|X]$ so for every input word $i_{k} \in \mathbb{F}^{k}$ we have a codeword $\bar{c}\in C$: $\bar{c} = [i_{k}|i_{k}\cdot X]$. For every $\bar{y}\in F^{n}$, $\bar{y} = [i_{k}|Y_{n-k}]$ where $Y_{n-k}\in \mathbb{F}^{n-k}$. So it can be easily viewed that for every $\bar{y} \exists \bar{c} : \min d(\bar{y},\bar{c}) \leq n-k$. And taking $\max$ over all gives covering radius.