(Note that in order to repair a super symbol, we can obtain the value of r out required super symbols by contacting (frac {r}{3}) symbols over (mathbb {F}_{q}) from each of their corresponding codewords of the inner code).
When the LDPC code is combined with an inner recursive code, as shown in Figure 3, these two degree-1 variable nodes will cause a minimum distance of for the entire codeword, irrespective of the code length.
Note that the SNR of the source to relay channel is always higher than the SNR of the source to destination channel as |h SR |2>|h SD |2, therefore the successful decoding of the codeword w1 of the code C 1 is possible at the relay even when it fails at the destination.
In particular, the supports of all the weight 8 codewords in the code form a Steiner system S 5,8,24).
Now the claim that (r leq ell Delta cdot text {rate}({mathcal C}_{0})) follows from the fact that correcting each erasure requires contacting at least (Delta cdot text {rate}({mathcal C}_{0})) code symbols from a codeword of the shorter code ({mathcal C}_{0}).
The obtained array is a codeword of the product code.
The ratio in (7) relates the possible decoder errors, i.e. cases with up to t(s) Hamming distance to a wrong codeword of the BCH code to the total number of possible syndromes.
An evaluation method for the new codes is presented; this considers intervals used for all codewords in each code set and offers a comparison with those of conventional codes.
The codewords of code C are assumed to have equally probable values of the encoded bits, i.e., the probability, Pr {ci=0}=Pr {ci=1}=1/2, for i=1,2,…,N.
Consider a binary linear code C j (n,k,d) in which n is the codeword length, k is the dimension of the code, and d is the minimum Hamming distance of C j.
