Question

I am currently working on convolutional codes in Matlab. One way to describe such a code is by its parity-check matrix H, which looks e.g. like this for the case of a R = b/c = 2/3 code:

[ 1   D   D^3 ]
[ D^3 D^2 1   ]

I would like to turn this matrix into systematic form, i.e. the first b x b columns should form an identity matrix. For the above example this would be something like:

[ 1   0   ? ]
[ 0   1   ? ]

My question is how can such a matrix, in which each entry is a polynomial, be represented in Matlab most conveniently? I was thinking of a matrix of coefficient vectors, but this seems kind of unwieldy. At the moment I just can’t figure out the best way to approach this problem without creating unnecessary complexity.

Some further remarks:
The coefficients are from GF(2) so all calculations are modulo 2 i.e. 1 + 1 = 0, but this should not be problematic after this question is answered.
General hints and gotchas concerning this topic will be highly appreciated 🙂

Question answered by EitanT, with some limitations on the polynomial degree(maximum 64, due to 64 bit precision).

Answer 1

Since the coefficients belong to GF(2), you can represent each polynomial as a binary number, where each bit represents the corresponding power. For example: D³+D² = 1100₂ = 12

This allows you to store H as a simple matrix and perform rather fast binary operators (such as XOR) when transforming it to reduced row echelon form to obtain the systematic form.

The H matrix in your example would look like this:

H = [1 2 8;
     8 4 1]