Abstract
Let p be a prime number and assume p ≥ 5. We will use a result of L. Redéi to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p − 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.
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References
Best M.R. (1983). Perfect codes hardly exist. IEEE Trans. Inform. Theory 29(3): 349–351
Blokhuis A., Lam C.W.H. (1984). More coverings by rook domains. J. Combin. Theory Ser. A 36: 240–244
Etzion T. (1996). Nonequivalent q-ary perfect codes. SIAM J. Discrete Math. 9(3): 413–423
Golay M.J.E.: Notes on digital coding. Proc. IRE 37 (1949), Correspondence 657.
Hamming, Error detecting and error correcting codes. Bell Syst. Tech. J. 29, 147–160 (1950).
Heden O. (1975). A generalised Lloyd Theorem and mixed perfect codes. Math. Scand. 37: 13–26
Heden O.: Perfect codes from the dual point of view I. submitted to Discrete Math.
Heden O. (2006). A full rank perfect code of length 31. Des. Codes Crypthogr. 38, 125–129
Lang S. (1965). Algebra. Addison-Wesley Publishing Company, Reading
Leontiev V.K., Zinoviev V.A. (1973). Nonexistence of perfect codes over galois fields. Problems Inform. Theory 2(2): 123–132
Lindström B. (1969). On group and nongroup perfect codes in q symbols. Math. Scand. 25: 149–158
Phelps K.T. (1983). A general product construction of perfect codes. SIAM J. Algebra Discrete Method. 4, 224–228
Phelps K.T., Villanueva M. (2002). Ranks of q-ary 1-perfect codes. Des. Codes Cryptogr. 27(1–2): 139–144
Phelps K.T., Rifà J., Villanueva M. (2005). Kernels and p-kernels of p r-ary 1-perfect codes. Des. Codes Cryptogr. 37(2): 243–261
Rédei L.: Lückenhafte Polynome über endlischen Körpern. Birkhäuser verlag Basel und Stuttgart (1970).
Schönheim J. (1968). On linear and nonlinear single-error-correcting q-ary perfect codes. Inform. Control 12: 23–26
Solov’eva F.I.: On Perfect Codes and Related Topics. Com2Mac Lecture Note Series 13, Pohang (2004).
Tietäväinen A. (1973). On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24, 88–96
van der Waerden B.L. (1966). Algebra, Erster Teil, Siebte Auflage der Modernen Algebra. Springer-Verlag, Berlin
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Communicated by V.A. Zinoviev.
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Heden, O. On perfect p-ary codes of length p + 1. Des. Codes Cryptogr. 46, 45–56 (2008). https://doi.org/10.1007/s10623-007-9133-y
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DOI: https://doi.org/10.1007/s10623-007-9133-y