bchdec

BCH-decode an input that has more errors per codeword than the error correcting capability of the BCH decoder. Decode a BCH coded message with two errors per codeword using a single-error correcting BCH decoder. View the effects of the error mismatch on the output codeword.

Check the number of errors per codeword a [63,57] BCH decoder is capable of correcting.

n = 63; 
k = 57;
t = bchnumerr(n,k)

t = 
1

The [63,57] BCH decoder is capable of correcting one error per codeword.

Create a random stream and use it to generate a GF array. Encode the message.

s = RandStream('swb2712','Seed',9973);
msg = gf(randi(s,[0 1],1,k));
code = bchenc(msg,n,k);

Add two errors per codeword and decode the errored code.

cnumerr2 = zeros(nchoosek(n,2),1);
nErrs = zeros(nchoosek(n,2),1);
cnumerrIdx = 1;
for idx1 = 1 : n-1
    %sprintf('idx1 for 2 errors = %d', idx1)
    for idx2 = idx1+1 : n
        errors = zeros(1,n);
        errors(idx1) = 1;
        errors(idx2) = 1;
        erroredCode = code + gf(errors);
        [decoded2, cnumerr2(cnumerrIdx)] ...
          = bchdec(erroredCode,n,k);

Encode the decoded message. Check that the re-encoded message differs from the errored message in only one bit.

        if cnumerr2(cnumerrIdx) == 1
            code2 = bchenc(decoded2,n,k);
            nErrs(cnumerrIdx) = biterr(double(erroredCode.x), ...
              double(code2.x));
        end        
        cnumerrIdx = cnumerrIdx + 1;    
    end
end

Plot the computed number of errors, based on the difference between the doubly-errored code and the re-encoded version of the initial decoding.

plot(nErrs)
title('Number of Actual Errors')

All inputs with two errors were decoded to a codeword that differs in exactly one bit from the re-encoded version.

Set the BCH parameters for a Galois array of GF(2).

M = 4;
n = 2^M-1;   % Codeword length
k = 5;       % Message length
nwords = 10; % Number of words to encode

Create a message.

msgTx = gf(randi([0 1],nwords,k));

Find the error-correction capability.

t = bchnumerr(n,k)

t = 
3

Encode the message.

enc = bchenc(msgTx,n,k);

Corrupt up to t bits in each codeword.

noisycode = enc + randerr(nwords,n,1:t);

Decode the noisy code.

msgRx = bchdec(noisycode,n,k);

Validate that the message was properly decoded.

isequal(msgTx,msgRx)

ans = logical
   1

Increase the number of possible errors, and generate another noisy codeword.

t2 = t + 1;
noisycode2 = enc + randerr(nwords,n,1:t2);

Decode the new received codeword.

[msgRx2,numerr] = bchdec(noisycode2,n,k);

Determine if the message was properly decoded by examining the number of corrected errors, numerr. Entries of -1 correspond to decoding failures, which occur when the codeword has more errors than can be corrected for the specified [n,k] pair.

numerr

numerr = 10×1

     1
     2
    -1
     2
     3
     1
    -1
    -1
     2
     3

Two of the ten transmitted codewords were not correctly received.