A family of rate 1/2 modified binary block repetition codes

We design a family of rate 1/2 linear binary block codes having low complexity of systematic encoding and decoding even for large block-length. The construction is derived from ordinary block repetition codes. The poor distance properties of ordinary block repetition codes are improved using modulo 2 sums of cyclic shifts of the input information vector. The minimum Hamming distance is a design parameter chosen according to the code block-length and the desired complexity of the encoder and decoder. The input-output weight enumerators of some of the proposed codes are obtained conditioned on knowledge of the input-output weight enumerators of the underlying codes. The input-output weight enumerators can be used to upper-bound the probability of decoded bit-error. We consider code lengthening, and it turns out that the perfect binary Golay code belongs to the family of the proposed lengthened modified binary block repetition codes. Three examples are used to study the error-correcting capabilities on Gaussian channels, and to describe the decoding algorithms. The results indicate that the proposed codes are viable alternatives to ordinary binary block repetition codes in low-complexity, low-power and low-cost applications.