For a linear non-binary code over a finite field, what are the ways of finding the minimum Hamming Distance. I can use the SAGEmath minimum_distance() command, but I'd like to be able to do it without software as well.
My understanding of the minimum distance of this code is the minimum number of places in which corresponding symbols are different between codewords.
I know that for binary linear codes, the minimum distance of a code is equal to the minimum hamming weight of the code. Does this hold for a non-binary code over a finite field?
For example, a code over GF(4) (which has field elements 0, 1, w, w+1), is generated by $$G = \begin{pmatrix} 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & w & w+1 & 1 \end{pmatrix}$$ has codewords
- 0,0,0,0,0
- 1,1,1,1,0
- w,w,w,w,0
- w+1,w+1,w+1,w+1,0
- 0,1,w,w+1,1
- 1,0,w+1,w,1
- w,w+1,0,1,1
- w+1,w,1,0,1
- 0,w,w+1,1,w
- 1,w+1,w,0,w
- w,0,1,w+1,w
- w+1,1,0,w,w
- 0,w+1,1,w,w+1
- 1,w,0,w+1,w+1
- w+1,0,w,1,w+1
- w,1,w+1,0,w+1
So if I was using the Hamming weight, the minimum weight of non-zero codewords in this code is clearly $4$. Can I use this for the minimum distance? Will this work for all linear codes over finite fields?
I have seen people encode each element of GF(4) as a binary strings (where non-zero codewords are encoded as binary strings of equal weight), but I am not sure what the need is for this unless we cannot find the Hamming distance in the above way.
So if the Hamming weight method is not possible over non binary finite fields, is there a way of finding minimum distance that does not rely on manually checking the distance between each codeword?
Any further reading/resources appreciated also.
Are you sure that your list of codewords is exhaustive? As, I didn't see the word $(1,w,0,w+1,0) = (w+1, w+1, w+1, w+1, w+1) +_2 (w, 1, w+1, 0, w+1)$ which has weight equal to 3 and then it clearly checks out the comment by @Lahtonen and you get the minimum distance equal to 3.