Abstract
The geometric codes are the duals of the codes defined by the designs associated with finite geometries. The latter are generalized Reed–Muller codes, but the geometric codes are, in general, not. We obtain values for the minimum weight of these codes in the binary case, using geometric constructions in the associated geometries, and the BCH bound from coding theory. Using Hamada's formula, we also show that the dimension of the dual of the code of a projective geometry design is a polynomial function in the dimension of the geometry.
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References
E. F. Assmus, Jr. and J. D. Key, Polynomial codes and finite geometries, Chapter 16, pp. 1269-1343, Handbook of Coding Theory (V. Pless and W. C. Huffman, eds.), Elsevier (1998).
E. F. Assmus, Jr. and J. D. Key, Designs and their Codes, Cambridge Tracts in Mathematics, Cambridge University Press, 103 (1992) (Second printing with corrections, 1993).
I. F. Blake and R. C. Mullin, The Mathematical Theory of Coding, Academic Press, New York (1975).
W. Bosma and J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of Sydney (November 1994).
A. E. Brouwer and H. A. Wilbrink, Block designs, Handbook of Incidence Geometry (F. Buekenhout, ed.), Elsevier (1995) chap. 8, pp. 349-382.
P. V. Ceccherini and J. W. P. Hirschfeld, The dimension of projective geometry codes, Discrete Math., Vol. 06,No. 107 (1992) pp. 117-126.
P. Delsarte, BCH bounds for a class of cyclic codes, SIAM J. Appl. Math., Vol. 19 (1970) pp. 420-429.
P. Delsarte, J. M. Goethals, and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives, Inform. and Control, Vol. 16 (1970) pp. 403-442.
P. Delsarte, A geometric approach to a class of cyclic codes, J. Combin. Theory, Vol. 6 (1969) pp. 340-358.
P. Delsarte, On cyclic codes that are invariant under the general linear group, IEEE Trans. Inform. Theory, Vol. 16 (1970) pp. 760-769.
D. G. Glynn and J. W. P. Hirschfeld, On the classification of geometric codes by polynomial functions, Des. Codes Cryptogr., Vol. 6 (1995) pp. 189-204.
J. M. Goethals and P. Delsarte, On a class of majority-logic decodable cyclic codes, IEEE Trans. Inform. Theory, Vol. 14 (1968) pp. 182-188.
N. Hamada, The rank of the incidence matrix of points and d-flats in finite geometries, J. Sci. Hiroshima Univ. Ser. A–I, Vol. 32 (1968) pp. 381-396.
N. Hamada, On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes, Hiroshima Math. J., Vol. 3 (1973) pp. 153-226.
J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford (1985).
J. W. P. Hirschfeld and X. Hubaut, Sets of even type in P G(3, 4) alias the binary (85,24) projective code, J. Combin. Theory Ser. A, Vol. 29 (1980) pp. 101-112.
J. W. P. Hirschfeld and R. Shaw, Projective geometry codes over prime fields, Finite Fields: Theory, Applications, and Algorithms, Contemporary Math., Amer. Math. Soc., Providence, 168 (1994) pp. 151-163. (Las Vegas, 1993).
S. Packer, On sets of odd type and caps in Galois geometries of order four, PhD thesis, University of Sussex, 1995.
S. Packer, On sets of odd type in P G(n, 4) with 21 Hermitian prime sections, Boll. Un. Mat. Ital. B, Vol. 11 (1997) pp. 203-225.
S. Packer, On sets of odd type in P G (4, 4) and the weight distribution of the binary [341,45] projective geometry code, J. Geom., to appear.
A. Pott, On abelian difference set codes, Des. Codes Cryptogr., Vol. 2 (1992) pp. 263-271.
B. F. Sherman, On sets with only odd secants in geometries over G F (4), J. London Math. Soc., Vol. 27 (1983) pp. 539-551.
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Calkin, N.J., Key, J.D. & Resmini, M.J.d. Minimum Weight and Dimension Formulas for Some Geometric Codes. Designs, Codes and Cryptography 17, 105–120 (1999). https://doi.org/10.1023/A:1008362706649
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DOI: https://doi.org/10.1023/A:1008362706649