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On optimal binary codes with unbalanced coordinates

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

A coordinate of a binary code of size M is said to be balanced if the number of zero and ones in the coordinate is either \(\lfloor M/2\rfloor \) or \(\lceil M/2 \rceil \) (that is, exactly \(M/2\) for even M). Since good codes (of various types) tend to be balanced in all coordinates, various conjectures have been made regarding the existence of such codes. It is here shown that there are parameters for which there are no optimal binary error-correcting codes with a balanced coordinate. This is proved by the code attaining \(A(17,8) = 36\), which is shown to be unique here; \(A(n,d)\) denotes the maximum size of a binary code of length n and minimum distance d. It is further shown that \(A(18,8) \le 68\).

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Acknowledgments

The author is grateful to Neil Sloane for encouraging him to try to find out whether an \((18,72,8)\) code exists or not. This work was supported in part by the Academy of Finland, Grant Number 132122.

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  1. Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 , Aalto, Finland

    Patric R. J. Östergård

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  1. Patric R. J. Östergård
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Correspondence to Patric R. J. Östergård.

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Östergård, P.R.J. On optimal binary codes with unbalanced coordinates. AAECC 24, 197–200 (2013). https://doi.org/10.1007/s00200-013-0189-9

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