Identically zero coordinate of a linear code

Question

In a book (Covering Codes by Cohen, Honkala, Litsyn, Lobstein) I found the statement that the covering radius of a linear code without identically zero coordinate is at most $\lfloor n / 2 \rfloor$. I would like to apply this statement on Simplex-Codes but I don't understand the term 'identically zero coordinate'. What does it refer to? I am not sure if I can apply the statement since every Simplex-Code contains the zero codeword.

Answer 1

I am fairly sure that identically zero coordinate (note the singular form) means that there exists a bit position, say the $i$th, such that the $i$th bit of every codeword is equal to zero.

The relation to covering radius is easy to understand. As an extreme example, if your code consists of only two words, $C=\{00000\ldots0, 10000\ldots0\}$, then its covering radius is $n-1$, because the minimum distance from the all $1$s vector to a codeword is $n-1$. Compare with the code $C=\{0000\ldots0,1111\ldots1\}$. Here the covering radius is $[n/2]$, because a given vector is at at most that Hamming distance from one of the words (use majority logic to find the closest match).

Anyway, the simplex code has no identically zero coordinate, so you can proceed.