On the Multiuser Error Probability and the Maximum Likelihood Performance of MDS Codes

MDS (e.g. Reed-Solomon) codes have many desirable properties which make them the code of choice in network scenarios and distributed coding schemes. An average binary weight enumerator of Reed-Solomon (RS) codes is derived assuming a binomial distribution of the bits in a non-zero symbol. Lower bounds on the average binary minimum distance of the ensemble of binary images of a Reed-Solomon code are shown. The ensemble of binary images of the RS code is shown to be, on average, asymptotically good. The performance of bit-level Reed-Solomon maximum likelihood decoders is studied. Given an arbitrary partition of the coordinates of a code, we introduce the partition weight enumerator which enumerates the codewords with a certain weight profile in the partitions. A closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived. Using this result, some properties of MDS codes are discussed. In particular, we show that all coordinates have the same weight within the subcodes of constant weight codewords. The results are extended for the ensemble of binary images of MDS codes defined over finite fields of characteristic two. The error probability of Reed-Solomon codes in multiuser networks is then studied. This research was supported by NSF grant no. CCF-0514881 and grants from Sony, Qualcomm, and the Lee Center for Advanced Networking. This work was presented in part at the 2004 42nd Allerton Conf. on Communication, Control and Computing [1] and at the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia [2].