In this note, we give a correction to the Proof of Theorem 2 in L. G. Tallini, S. Al-Bassam, and B. Bose, ldquoFeedback Codes Achieving the Capacity of the Z-Channel,rdquoIEEETransactionsonInformationTheory, vol. 54, pp. 1357-1362, March 2008.

Given the 1 to 0 bit error probability, pisin[0, 1], the capacity of the Z-channel is given by Cz=log2(1+pp/(1-p)-p1/(1-p)). Some new error free feedback coding schemes that achieve the Z-channel capacity are presented.

Let /spl Phi//sub m//spl sube/ /spl Copf/ be the set of all mth roots of unity, m/spl isin/ IN. A balanced code over /spl Phi//sub m/ is a block code over the alphabet /spl Phi//sub m/ such that each code word is balanced; that is, the complex sum of its components (or weight) is equal to 0. Let B/sub m/(n) be the set of all balanced words of length n over /spl Phi//sub m/. In this correspondence, it is shown that when m is a prime number, the set B/sub m/(n) is not empty if, and only if, m divides n. In this case, the minimum redundancy for a balanced code over /spl Phi//sub m/ of length n is. On the other hand, it is shown that when m=4, the set B/sub 4/(n) is not empty if, and only if, n is even, and in this case, the minimum redundancy for a balanced code over /spl Phi//sub 4/ of length n is. Further, this correspondence completely solves the problem of designing efficient coding methods for balanced codes over /spl Phi//sub m/, when m=4. In fact, it reduces the problem of designing efficient coding schemes for balanced codes over /spl Phi//sub 4/ to the design of efficient balanced codes over the usual bipolar alphabet /spl Phi//sub 2/={-1,+1}.

An improved method to construct t-EC/AUED (t error-correcting and all unidirectional error detecting codes) is presented. In some cases, the new method improves the best known t-EC/AUED codes. Furthermore, the construction does not require complementation of codewords except the all zero and the all ones words.

A new single asymmetric error-correcting code is proposed. This code is constructed using a product of two codes of smaller dimensions. The proposed code is of dimension 17 and of size 8192, i.e., with 2/sup 13/ codewords. The best known code of dimension 17 has size 7968 and capable of handling 12 information bits only. The only other three known cases of single asymmetric codes accommodating one more extra information bit than the symmetric case are for code dimensions 2, 4, and 16.

It is proved that for all values of code length n, except when n=2, 4, and 8 and possibly when n=2/sup r/ and n=2/sup r/+1, where r/spl ges/1, the Hamming codes are also optimal systematic single asymmetric error-correcting codes. For the cases n=2/sup r/ and n=2/sup r/+1, r/spl ges/4, when not all information words are used, two efficient systematic 1-asymmetric codes are described.

New single asymmetric error-correcting codes are proposed. These codes are better than existing codes when the code length n is greater than 10, except for n=12 and n=15. In many cases one can construct a code C containing at least [2/sup n//n] codewords. It is known that a code with |C|/spl ges/[2/sup n//(n+1)] can be easily obtained. It should be noted that the proposed codes for n=12 and n=15 are also the best known codes that can be explicitly constructed, since the best of the existing codes for these values of n are based on combinatorial arguments. Useful partitions of binary vectors are also presented.

All words in a balanced code have equal number of ones and zeros. Denote by DC(n,k) a balanced (or dc-free) code of length n, and 2/sup k/ code words. We design an efficient DC(k+r, k) code with k=2/sup r+1//spl minus/0.8/spl radic/(r/spl minus/2). These codes are optimal up to the construction method, introduced by D.E. Knuth (1986).>

Introduces the theory and design of codes that correct t asymmetric errors and simultaneously detect d(d>t) asymmetric errors (t-AEC/d-AED). These codes have a much higher information rate than symmetric error correcting/detecting codes and yet maintain the same encoding and decoding complexity as existing error correcting codes. They are suited for asymmetric channels such as digital transmission over optical fiber or metallic cable and recording of data on optical disks. The authors also improve the design of the best-known t-EC/AUED (t symmetric error correcting and all unidirectional error detecting) codes and t-EC/d-UED (t symmetric error correcting and d unidirectional error detecting) codes.>

New constructions of t-error correcting balanced codes, for 1>

Efficient byte unidirectional error correcting codes that are better than byte symmetric error correcting codes are presented. The encoding and decoding algorithms are discussed. A lower bound on the number of check bits for byte unidirectional error correcting codes is derived. It is then shown that these codes are close to optimal. Capability of these codes for asymmetric error correction is also described. Codes capable of detecting double byte unidirectional errors are also given.>

In a balanced code each codeword contains equally many 1's and 0's. Parallel decoding balanced codes with 2/sup r/ (or 2/sup r/-1) information bits are presented, where r is the number of check bits. The 2/sup 2/-r-1 construction given by D.E. Knuth (ibid., vol.32, no.1, p.51-3, 1986) is improved. The new codes are shown to be optimal when Knuth's complementation method is used.>