Abstract
For any prime power q we construct a nonlinear code consisting of 8q codewords, each codeword having length 4q, and with minimum distance 2(q − 1). When compared to a Hadamard code of the same size it is shown that the new code corrects at most one error less, irrespective of q. Hadamard codes, however, are not known to always exist, whereas the new codes exist always.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berndt B.C., Evans R.J., Williams K.S.: Gauss and Jacobi Sums. John Wiley & Sons, New York (1998)
de Launey, W., Flannery, D.: Algebraic Design Theory. Mathematical Surveys and Monographs, Vol. 175. Amer. Math. Soc., Providence, RI (2011)
Dickson, L.E.: Cyclotomy, higher congruences, and Waring’s problem. Amer. J. Math. 57, 391–424, 463–474 (1935)
Goethals J.-M., Seidel J.J.: Orthogonal matrices with zero diagonal. Canad. J. Math. 19, 1001–1010 (1967)
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. Elsevier/North-Holland, Amsterdam (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
Funded under NIH grants P50-GM-53789, RO1-HL-076157 and an IBM shared University Research Award.
Rights and permissions
About this article
Cite this article
Constantine, G.M. An Alternative to Hadamard Codes — One Error for the Price of Existence. Ann. Comb. 19, 421–425 (2015). https://doi.org/10.1007/s00026-015-0274-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-015-0274-9