How to generate all possible SEC matrices for $n$-bit

Question

I want to know if there is an algorithm with which man can generate all possible forms of single error correction matrices for n bits of data, both linear and non-linear. For example all possible $32$-bit single error ($d=3$) correction matrices.

Answer 1

This may be easier to think about in terms of parity check matrices. In order to get $d=3$ you need a parity check matrix $H$ with the following two conditions:

1. All the columns must be non-zero.
2. All the columns must be distinct.

You need 32 such columns, so they must have at least 6 entries each, IOW the matrix $H$ must have at least six rows.

Because you need to avoid repetitions of codes you also need to have mechanism for avoiding duplicates. To that end we need to make the observation that the two check matrices, say $H$ and $H'$, yield the same code if and only if they can be gotten from each other by a sequence of elementary row transformations. From linear algebra you recall that among the set of row equivalent matrices there is a unique matrix in the reduced row echelon form. This is the third constraint. Of course we can (should) drop out any all zero rows from $H$ because they are useless as check equations.

You need to list $k\times 32$ binary matrices $H$, $k\ge6$, such that $H$ is in row echelon form, has no columns of all zeros and no repeated columns.