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On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

Abstract

It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length \(2q+2\) contains the (0,1)-incidence matrix of a Hadamard 3-\((2q+2,q+1,(q-1)/2)\) design D(q) associated with a Paley–Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley–Hadamard matrix of type I. If \(q=5, 11, 17, 23\), then the full permutation automorphism group of L(q) coincides with the full automorphism group of D(q), and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code C(17) are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley–Hadamard matrix H of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix \(H'\) such that the symmetric 2-(36, 15, 6) design D associated with \(H'\) has trivial full automorphism group, and the incidence matrix of D spans a ternary code equivalent to C(17).

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Notes

  1. The symmetry code for \(q=5\) is equivalent to the extended ternary Golay code.

  2. This is the order of the Paley–Hadamard matrix of Type II for \(q=17\) [7].

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Acknowledgements

The author thanks Cary Huffman for reading a preliminary version of this paper and making several useful suggestions that led to an improvement of the text.

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Correspondence to Vladimir D. Tonchev.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”

Appendix

Appendix

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A 2-(36, 15, 6) design associated with the Pless symmetry code of length 36

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Tonchev, V.D. On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs. Des. Codes Cryptogr. (2021). https://doi.org/10.1007/s10623-021-00941-0

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Keywords

  • Pless symmetry code
  • Hadamard matrix
  • Hadamard 3-design
  • Hadamard 2-design
  • Paley–Hadamard matrix

Mathematics Subject Classification

  • 05B05
  • 05B20
  • 94B05