Decyclic codes

Conclusion

We will compare the parameters of binary decyclic codes with the corresponding parameters of binary Bose-Chaudhuri codes.

It is obvious that binary (n+1, k) codes have better redundancy parameters than Bose-Chaudhuri codes with the same minimal code distance. Thus, in [3] it was shown that among binary cyclic codes of length n>4 there are no equivalents to the optimal Hamming binary code with minimal code distance d=4. (The authors of [3] regard as optimal a linear (n.k) code with n-k check symbols and a minimal Hamming distance of d, if it has the minimum possible length n for given d and k). A decyclic code of length n+1=2m, with d=4, will be similar to the Hamming code. In Table 2 the parameters of some decyclic (n+1+c, k) codes are listed together with the parameters of the corresponding cyclic Bose-Chaudhuri codes [2]. It can be seen from this table that, for the examples given, decyclic codes are more optimal than the corresponding Bose-Chaudhuri codes.

Thus, the decyclic codes constructed have the following properties.

• 1. They can be formed by a constructive method.

• 2. In the majority of cases, decyclic codes have better redundancy parameters than the corresponding Bose-Chaudhuri codes.

• 3. The decoding process is comparable in complexity to that of the Bose-Chaudhuri codes.