# Equivalent codes

I'm having problems by understanding the use one can do of equivalent codes. To solve problems about a linear code with a generator matrix G, can I always assume that the matrix is in systematic form (ie (id|R))?

Which are the features of equivalent codes in general? Thanks

• Any linear code is equivalent to a code which has a generator matrix in systematic form, but not every linear code has a generator matrix in systematic form. Does this help? – Git Gud Jan 19 '15 at 12:13
• May be... So if I want to prove something about any (n,k) linear codes, can I always assume that it's generator is in systematic form? – sky90 Jan 19 '15 at 12:33
• No, equivalence of codes doesn't preserve everything, but it preserves a lot of interesting things like length, dimension, size and minimum distance. – Git Gud Jan 19 '15 at 12:36
• Agree with Git Gud. – Jyrki Lahtonen Jan 20 '15 at 18:19

## 1 Answer

Warning : This answer is wrong, it assumes a too restricted (almost trivial) concept of equivalent codes. See Jyrki Lahtonen's comment below. I'll delete it as soon as it's un-accepted.

To add to GitGud's comments: the most relevant common feature of two equivalent codes is that they have the same codebook (set of codewords, which is the row space of $G$).

Hence, any property of some linear code $c_1$ with generator matrix $G_1$ will be shared by another equivalent code $c_2$ with generator matrix $G_2$, as long as that property is inherent to the codebook alone (for example: minimum distance).

A property that is not preserved by equivalent codes is the mapping of raw inputs to codewords - of course, elsewhere the two codes would be not just equivalent but identical.

• I disagree. The last time I looked it up two (binary) codes where defined to be equivalent, iff their respective codebooks can be gotten from each other by permuting the bit positions. So for example the matrices $$G_1=\pmatrix{1&1&0&0&\cr0&0&1&1\cr}$$ and $$G_2=\pmatrix{1&0&1&0&\cr0&1&0&1\cr}$$ generate equivalent codes. You get them from each other by swapping the second and third bits of all the codewords. Equivalent codes share the same minimum distance, and the same weight distributions. However, they may have different trellis complexities - important when (soft) decoding. – Jyrki Lahtonen Jan 20 '15 at 18:18
• @JyrkiLahtonen My bad, answer updated. – leonbloy Jan 20 '15 at 18:55