A low-rate improvement on the Elias bound (Corresp.)

RU4LEtWH&!4sl-A'ERG RU4LEtWH&!4sl Fig. 1. Fig. 1. Entropy per cell as function of average run length. Entropy per cell as function of average run length. distributed: for tl 2 0 for tl < 0 where t, is the (continuous) run length. The average run length is f, = 6. (20) Now, suppose the continuous run length tl is quantized to obtain the discrete run length t : t = [t1] + 1 (21) where [tl] = largest integer in t,. (23) 1 t=-. 1-ebb Letting .? = a, we obtain 1 a=-1-e-* or b = log a. a-l Substituting (26) into (23), we obtain (25) (26) 1 qi=-e-Cloga-Iog(a-1)11 a-l which is identical to (15). Therefore, the quantized Poisson square wave achieves the maximum entropy given by (16). (28) After the aforementioned analysis was performed, it was found that the quantized Poisson square wave is identical to Capon's first-order Markov chain model [l 1, if we set P(0 IO) = P(1 1 1) = 9. Therefore, although Capon apparently did not realize it, the saving in bits predicted by his model is actually a lower bound for any two-level source with average white run length l/ (1-P(l 1 1)) and average black run length l/(1-P(0 IO)), because the run lengths in Capon's model are independent, and the exponential distributions of the white run lengths and the black run lengths ensure that both achieve the maximum entropy. REFERENCES [l] J. Capon, " A probabilistic model for run-length coding of pictures, " IRE Trans. Absrrucr-An upper bound on the minimum distance of binary blocks codes, which is superior to Elias' bound for R < 0.0509+, is obtained. The new bound has the same derivative (co) at R = 0 as Gilbert's lower bound. (Elias' bound has derivative-In 2 at R = 0). For R between 0 and 1 denote by d(n,R) the largest possible minimum distance for a binary block code of length n and rate 2 R. It is unknown whether D(R) = lim i d(n,R) n-rm I1 exists, so let us define D(R) = lim sup 1 d(n,R) n-bm n e(R) = lim inf 1 d(n,R). n-too n Until now the best bounds on D(R) and p(R) have been L!(R) 2 fU-RI