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# Extension of Knuth's Balancing Algorithm with Error Correction

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Knuth's celebrated balancing method consists of inverting the first z bits in a binary information sequence, such that the resulting sequence has as many ones as zeroes, and communicating the index z to the receiver through a short balanced prefix. In the proposed method, Knuth's scheme is extended with error-correcting capabilities, where it is allowed to give unequal protection levels to the prefix and the payload. An analysis with respect to the redundancy of the proposed method is performed, showing good results while maintaining the simplicity features of the original scheme.
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Knuth's celebrated balancing method consists of inverting the first bits in a binary information sequence, such that the resulting sequence has as many ones as zeroes, and communicating the index to the receiver through a short balanced prefix. In the proposed method, Knuth's scheme is extended with error-correcting capabilities, where it is allowed to give unequal protection levels to the prefix and the payload. The proposed scheme is very general in the sense that any error-correcting block code may be used for the protection of the payload. Analyses with respect to redundancy and block and bit error probabilities are performed, showing good results while maintaining the simplicity features of the original scheme. It is shown that the Hamming distance of the code is of minor importance with respect to the error probability.
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Let n be an even positive integer and F be the field \GF(2). A word in F^n is called balanced if its Hamming weight is n/2. A subset C \subseteq F^n\$ is called a balancing set if for every word y \in F^n there is a word x \in C such that y + x is balanced. It is shown that most linear subspaces of F^n of dimension slightly larger than 3/2\log_2(n) are balancing sets. A generalization of this result to linear subspaces that are almost balancing'' is also presented. On the other hand, it is shown that the problem of deciding whether a given set of vectors in F^n spans a balancing set, is NP-hard. An application of linear balancing sets is presented for designing efficient error-correcting coding schemes in which the codewords are balanced. Comment: The abstract of this paper appeared in the proc. of 2009 International Symposium on Information Theory
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Let A(n,d,w) denote the maximum possible number of codewords in an (n,d,w) constant-weight binary code. We improve upon the best known upper bounds on A(n,d,w) in numerous instances for n&les;24 and d&les;12, which is the parameter range of existing tables. Most improvements occur for d=8, 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n&les;28 and d&les;14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n,d,w) by means of mapping constant-weight codes into Euclidean space. This approach produces, among other results, a bound on A(n,d,w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doubly-constant-weight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doubly-bounded-weight codes, which may be thought of as a generalization of the doubly-constant-weight codes. Subsequently, a class of Euclidean-space codes, called zonal codes, is introduced, and a bound on the size of such codes is established. This is used to derive bounds for doubly-bounded-weight codes, which are in turn used to derive bounds on A(n,d,w). We also develop a universal method to establish constraints that augment the Delsarte inequalities for constant-weight codes, used in the linear programming bound. In addition, we present a detailed survey of known upper bounds for constant-weight codes, and sharpen these bounds in several cases. All these bounds, along with all known dependencies among them, are then combined in a coherent framework that is amenable to analysis by computer. This improves the bounds on A(n,d,w) even further for a large number of instances of n, d, and w
Schouhamer Immink, \Knuth's Balanced Code Revisited
• J H Weber
J.H. Weber and K.A. Schouhamer Immink, \Knuth's Balanced Code Revisited", IEEE Trans. Inform. Theory, vol. 56, no. 4, pp. 1673-1679, April 2010.
• J H Weber
• K A Schouhamer Immink
• H C Ferreira
J.H. Weber, K.A. Schouhamer Immink, and H.C. Ferreira, \Error-Correcting Balanced Knuth Codes", submitted to IEEE Trans. Inform. Theory, January 2011.