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This paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets. Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upper bounds on minimax regret and lower bounds on minimax redundancy for such source classes. The general upper bounds emphasize the role of the normalized maximum likelihood (NML) codes with respect to minimax regret in the infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky—Trofimov coders for sources over finite alphabets. Up to logarithmic (resp., constant) factors the bounds are matching for source classes defined by algebraically declining (resp., exponentially vanishing) envelopes. Effective and (almost) adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes. Those results extend our knowledge concerning universal coding to contexts where the key tools from parametric inference are known to fail. [ABSTRACT FROM AUTHOR]

Let F be a finite field and b be a positive integer. A construction is presented of codes over the alphabet Fb with the following three properties: i) the codes are maximum-distance separable (MDS) over Fb, ii) they are linear over F, and iii) they have systematic generator and parity-cheek matrices over F with the smallest possible number of nonzero entries. Furthermore, for the case F = GF(2), the construction is the longest possible among all codes that satisfy properties i)–iii). [ABSTRACT FROM AUTHOR]