In this way a unified approach to weak universal block source coding is obtained. For the noiseless variable-rate coding and the fixed- rate coding with respect to ...
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applies. Thus Since q3 < l/3, this implies r’ < 1. Finally, assume q’=O, and let k’ be the highest number first-level node for which second-level nodes exist. For k P2>.. . }. Supposep E A, H(p) < cc. Then for some n, of fixed-rate codes for coding with respect to a single- E~l[&(Xn)] < cc. Since u,(X,) < Z[&(Xn)], it follows that letter fidelity criterion, obtaining results of Neuhoff, Gray, - E,, log p,W,> < 00. and Davisson [2] as special cases.In Theorem 4 a weakly Example: Let A be countably infinite. Let A be the universal sequenceof variable-rate codes for coding with family of all alphabet A ergodic sources with finite enrespect to a single-letter fidelity criterion is obtained, tropy. W e will show that the condition of L e m m a 1 is not which gives a strengthening of a result of Mackenthun satisfied, and thus there is not a universal weakly m inimax and Pursley [5, th. 31. sequenceof codes for A. Suppose there is a countable This paper provides a unified approach to the weak class 9 = {pI,p2;. . } satisfying the condition of L e m m a type of universal coding. The method of code construc- 1. W e may supposeA={1,2;..}. If p,EC?, then -log tion described above is not used to obtain a strongly pj(n)-+oo as n+co. Pick increasing sequences{x,o)}~=,, m inimax sequenceof codes in the senseof [l] and [2], nor {x~“}&*. from A such that no two sequenceshave a strongly universal sequence of codes as in [5]. The members in common. For each fixed j, pick a subresults that have been obtained in [l], [2], and [5] for this sequence{ xy)} r= i of {xi”)} YE, such that - log pj(x$ > strong type of universal coding impose restrictions on the 2”, n=l, 2,v.e. Let p0 be the PMF on A such that alphabet and on the class of sources in order that the pElo(xp) =2-n-j, proofs may be carried out. Indeed, the negative result nj= 1,2; * * given by [2, th. 6.51shows that some sort of restriction is for all other elementsa of A. h(a) = 0, necessary. Thus results of this stronger type, although more desirable, do not appear to be amenable to a unified Let p be the ergodic source on &” such that the family of projections {Xi} are independent under p and p[X, = a] = approach.
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&a), aEA. Then H(y)can be chosen so that ~~~E~ln-ll[~~(X~)]-H(~)I=O,
PEA, H(p) 0, where H( p) < cc. We show that for some n there exists a u’E S,, such that n-‘E,u’(X”) < H(p) + l . By noiselesscoding theory [6, problem 3.71there exists n >4/e for which there is a length function u of order n such that n-‘E,u(X”) 0: there is a countable set % of block sequence{B,,} T=, such that codes such that for any p E A, e > 0, and block code B, a) B,, is a block code of order n and r(B,J < R, n = there exists B’E % satisfying r(B’)if p is ergodic, b) implies that if p E A and code. W e find B’ E Ci3so that r(B’) CXEB 2-nR=2-nRIB,I. D,(R ‘), and r(BnR’)
