We give a complete classification of the error locator polynomials that occur in the Berlekamp decoding of double error correcting (DEC) Bose-Chaudhuri-Hocquenghem (BCH) codes. We present a new construction showing that all quadratic error locator polynomials produced by received vectors falling in the interstitial region between decoding spheres are illegitimate and have no roots. Furthermore, we show that a small subset of received vectors in the interstitial region produce cubic error locator polynomials that are illegitimate except for the correctable case of a triple error pattern with three equally spaced errors in the cyclic sense.

The author presents uncoded and coded performance results for noncoherent M-ary frequency-shift keying (MFSK) and differentially coherent binary phase-shift keying (DPSK) in a slow nonselective Nakagami-m (1960) fading channel. He gives simple expressions for the asymptotic slopes of probability of bit error for large signal-to-noise ratio and shows that the effective order of diversity compared to an uncoded Rayleigh channel is the product of two parameters, one for the channel and one for the code. He also compares the uncoded Nakagami-m results to those of the Rician channel in order to show performance differences between these two generalized fading channel models.>

For frequency-hopped (noncoherent) binary frequency shift keying (FH/BFSK) on a worst-case partial-band Gaussian interference channel, the bit error probability results are well known for the extreme cases where the signal is either nonfading or Rayleigh fading. In this work, the region between these extremes is filled in by considering the general Nakagami-m fading model. The worst-case partial-band Gaussian interference results are given by a one-parameter family which for m to infinity gives the Viterbi-Jacobs nonfading result, and for m=1 gives the Rayleigh fading result. In the latter case, a broadband interference strategy is optimal. Thus, the Nakagami-m results provide a smooth one-parameter bridge between the Viterbi-Jacobs channel and the Rayleigh fading channel. The results show that the worst-case interference fraction rho increases as the fading variance increases, up to Rayleigh fading. Any fading less severe than Rayleigh, however slight the departure from Rayleigh, requires a partial-band strategy for sufficiently large E/sub b//N/sub I/.>

The asymptotic (M to infinity ) probability of symbol error P/sub e/,/sub m/ for M-ary orthogonal modulation in a Nakagami-m fading channel is given by the incomplete gamma function P(m, mx) where x=In 2/(E/sub b//N/sub 0/) and E/sub b/ is the average energy per bit. For large signal-to-noise ratio this leads to a channel where the probability of symbol error varies as the inverse mth power of E/sub b//N/sub 0/. These channels exist for all m>or=1/2. The special case of m=1 corresponds to Rayleigh fading, an inverse linear channel.>

Application of the union bound to MFSK signaling on certain channels where probability of bit error varies in an inverse linear fashion withE_{b}/N_{0}can lead to the erroneous conclusion that performance degrades with increasing alphabet size. The anomaly is shown for the worst-case partial band Gaussian interference channel and the Rayleigh fading channel.