Arbitrary Alphabet Research Articles

Perfect codes hardly exist

The main lines of a partial proof of the "perfect code theorem" are presented. The relevant part refers to the nonexistence of unknown <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> -perfect codes over arbitrary alphabets for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t \not \in \{1, 2, 6, 8\}</tex> . The details of the proof can be found in the author's Ph.D. dissertation at the University of Amsterdam.

Maximal chains of subwords and up-down sequences of permutations

The free monoid generated by two letters is a poset for the relation “ u is a subword of v .” Two bijections between permutations and maximal chains starting from the empty word, involving the classical up-down sequence and the so-called odd-even height sequence, are given. For arbitrary alphabet, symmetric polynomials in s variables appear, generalizing the Eulerian polynomials and refining polynomials studied by Carlitz, Gessel, Riordan, and Stanley. The skewhook property of Foulkes is extended.

Extension of source coding theorems for block codes to sliding-block codes

For arbitrary alphabets and single-letter fidelity criteria, two theorems are given which allow any fixed-rate or variable-rate source coding theorem for block codes to be extended to sliding-block codes. Applications are given to universal coding and to the coding of a stationary nonergodic source.

Lee metric codes over integer residue rings (Corresp.)

A method for constructing Lee metric codes over arbitrary alphabet sizes using the elementary concepts of module theory is presented. The codes possess a high degree of symmetry. Codes with two information symbols over arbitrary alphabet sizes are cyclic reversible. For alphabet sizes which are a power of two or an odd prime number, codes with one information symbol are reversible and equidistant, and codes having more than two information symbols are quasi-cyclic reversible. Binary Reed-Muller codes arise as subcodes of the codes presented. A method of constructing equidistant Lee metric codes analogous to maximum length shift register codes is presented.

Composition bounds for channel block codes

The performance of Channel block codes for a general channel is studied by examining the relationship between the rate of a code, the joint composition of pairs of codewords, and the probability of decoding error. At fixed rate, lower bounds and upper bounds, both on minimum Bhattacharyya distance between codewords and on minimum equivocation distance between codewords, are derived. These bounds resemble, respectively, the Gilbert and the Elias bounds on the minimum Hamming distance between codewords. For a certain large class of channels, a lower bound on probability of decoding error for low-rate channel codes is derived as a consequence of the upper bound on Bhattacharyya distance. This bound is always asymptotically tight at zero rate. Further, for some channels, it is asymptotically tighter than the straight line bound at low rates. Also studied is the relationship between the bounds on codeword composition for arbitrary alphabets and the expurgated bound for arbitrary channels having zero error capacity equal to zero. In particular, it is shown that the expurgated reliability-rate function for blocks of letters is achieved by a product distribution whenever it is achieved by a block probability distribution with strictly positive components.

On Isomorphisms and Density of $NP$ and Other Complete Sets

If all $NP$ complete sets are isomorphic under deterministic polynomial time mappings (p-isomorphic) then $P \ne NP$ and if all PTAPE complete sets are P-isomorphic then $P \ne {\text{PTAPE}}$. We show that all $NP$ complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets. This shows that, in spite of the radically different origins and attempted simplification of these sets, all the known $NP$ complete sets are identical but for simple isomorphic codings computable in deterministic polynomial time.Furthermore, if all $NP$ complete sets are p-isomorphic then they all must have similar densities and, for example, no language over a single letter alphabet can be $NP$ complete, nor can any sparse language over an arbitrary alphabet be $NP$ complete. We show that complete sets in EXPTIME and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet. Similarly, we show that the hardest context-sensitive languages cannot be sparse. We als...

Systematic Construction of Cyclically Permutable Code Words

A numerical procedure for determining the cyclically permutable code of any length over an arbitrary alphabet is outlined. This method is easily programmed on a digital computer. A formula for the Hamming weight distribution of the code is also given. The results for the 12 bit binary cyclically permutable code are presented along with the weight distribution of the binary codes through length 14.

Statistical reliability synthesis of automata with arbitrary alphabets

Statistical reliability synthesis of automata with arbitrary alphabets

The capacity of the product channel

Given any pair of arbitrary alphabet channels satisfying coding theorem and its strong converse. Then the coding theorem and its strong converse also holds for the product channel and the capacity of the product channel equals the sum of the capacities of the components. Wyner derives this result in (Wyner, 1966) by means of list decoding methods (which obviously have no strong connection with this kind of problem). Here a simpler proof of the statement above is presented which uses only ‘maximal code estimate’ (see Wolfowitz, 1964) and a converse estimate.