decoding systematic Reed-muller code

Question

I work with a team, and we tried to build a an error correcting code. After going through some algorithm, we decided to start with the Reed-Muller code.

The coding is not complex, and we have managed to decode using MLD (majority logic decoding). Then we had a new constraint that the code must be systematic (the original message must be visible in the coded message). I transformed the generator matrix of Reed-Muller into systematic form.

I ran into difficulties when decoding.

My question is: can you use majorty logic decoding for systematic Reed-Muller? If yes, then how?

Second: is it worth going with the Reed-Muller algorithm for systematic codes or are there better systematic block codes (that are easy to decode).

Answer 1

Unless you did column interchanges while doing the reduction of the standard generator matrix $\mathbf G$ into reduced echelon form $\hat{\mathbf G} = [\mathbf I \mid P]$ to get the systematic version of the Reed-Muller code, the codewords of the systematic version of the Reed-Muller code are the same as the codewords of the standard version. (If you did column interchanges, undo them).

1. Now, decode the (possibly modified by column interchanges) "received word" that you thus formed to recover the "information bits" corresponding to the codeword that is closest to the modified received word.
2. Reconstruct the codeword by applying the encoding algorithm to the "information bits" determined in Step 1.
3. (Redo the column interchanges, if any)
4. What you have at this point is the most likely codeword (of the systematic Reed-Muller code) that was transmitted, and the actual information bits are "in the clear" just as you need them.

We cannot tell you any more because we have no information regarding whether you did any column interchanges or what the original Reed-Muller code generator matrix you began with; you need to work on this stuff yourself.

With regard to your other question re any better codes, that is a vast topic not suitable for this forum.