Posted by: atri | February 3, 2012

## Lect 8: More on Linear Codes

Today we looked at some properties of linear subspace and their implications for linear codes. The material is from Sec 2.2 and 2.3 in the book (which includes some of the proofs we skipped in class).

## Responses

1. Hello Professor,

According to theorem 2.2.6 in book,

If S is the proper subset of Fq, then |S| = q^k for some k >= 0. The parameter k is called the dimension of S.

What exactly is the dimension of a subspace? and how does it differ from ‘n’ of a finite field?does |S| imply the number of vectors in subspace S?

• Hi Harish,

I’m not sure what you mean by $n$ of a finite filed– maybe you’re talking about the set of vectors $\mathbb{F}_q^n$?

The dimension of a subspace $S\subseteq \mathbb{F}_q^n$ is the same as that of the corresponding linear code: $k=\log_q{|S|}$, where as you point out $|S|$ is the number of vectors in $S$. In general $0\le k\le n$ but it need not always be the case that $k=n$.

Let me know if you still have questions– in fact for interesting linear codes we will have $k.