We will show that • the entropy for a random variable gives a lower bound on the number of bits needed per character for a binary coding • Huffman codes are optimal in the average number of bits used per character among binary codes • the average bits per character used by Huffman codes is close to the entropy of the underlying random variable • one can get arbitrarily close to the entropy of a...

If pi (i = 1, . . . , N) is the probability of the i-th letter of a memoryless source, the length li of the corresponding binary Huffman codeword can be very different from the value − log pi. For a typical letter, however, li ≈ − log pi. More precisely, P m = ∑ j∈{i|li<− log pi−m} pj < 2 −m and P m = ∑ j∈{i|li>− log pi+m} pj < 2 −c(m−2)+2

In the present communication, we have obtained the optimum probability distribution with which the messages should be delivered so that the average redundancy of the source is minimized. Here, we have taken the case of various generalized mean codeword lengths. Moreover, the upper bound to these codeword lengths has been found for the case of Huffman encoding.

As is well known, Huffman’s algorithm is a remarkably simple, and wonderfully illustrative example of how to use the greedy method to design algorithms. However, the Huffman coding problem, which is to find an optimal binary character code (or an optimal binary tree with weighted leaves) is intrinsically technical, and its specification is ill-suited for students with modest mathematical sophis...

This paper studies the equivalence problem for cyclic codes of length p and quasi-cyclic codes of length pl. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183–215, 1993), who considered the special case p. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtai...

We consider the following variant of Huffman coding in which the costs of the letters, rather than the probabilities of the words, are non-uniform: “Given an alphabet of r letters of nonuniform length, find a minimum-average-length prefix-free set of n codewords over the alphabet;” equivalently, “Find an optimal r-ary search tree with n leaves, where each leaf is accessed with equal probability...

Černý’s conjecture asserts the existence of a synchronizing word of length at most (n− 1) for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p·a = q ·a for s...

The generic Huffman-Encoding Problem of finding a minimum cost prefix-free code is almost completely understood. There still exist many variants of this problem which are not as well understood, though. One such variant, requiring that each of the codewords ends with a “1,” has recently been introduced in the literature with the best algorithms known for finding such codes running in exponentia...

Some data compression techniques use large numbers of prefix-free codes. The following two techniques do so: adaptive Huffman encoding and bit recycling. Adaptive Huffman encoding allows successive symbols to be encoded where each one is encoded according to the statistics of the symbols seen so far. Bit recycling, on the other hand, is a technique that is designed to improve the efficiency of ...

−In this paper, the tolerance of Huffman Coding to memory faults is considered. Many pointer-based and array-based data structures are highly nonresilient to faults. A single fault in a memory array or atree node may result in loss of entire data or an incorrect code stream. In this paper, a fault tolerant designscheme is developed to protect the JPEG image compression system.