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A family of weakly self-dual codes

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Part of the Lecture Notes in Mathematics book series (LNM,volume 748)

Abstract

This note looks at a family of weakly self-dual codes, C, over GF(3), with (m×2m) generating matrix

, where J is the (m×m) matrix with every entry being +1, I the (m×m) identity matrix, and A the circulant incidence matrix of the (4t−1, 2t−1, t−1) block design, with first row being formed from the quadratic residues, mod(4t−1) (so m=4t−1, and is prime).

In particular, we consider these codes for t≡2 (mod 3) which is a necessary and sufficient condition for

.

In this case, we show C is contained within the code generated by

where H is the [(m+1)×(m+1)] Hadamard matrix. However, it is only by careful reduction of the length and dimension of F that allows us to reach a doubly circulant generating matrix.

We show the upper bound for the minimum distance for these codes is 2t+2. The minimum distance, d, for the first two cases of this family of (8t−2, 4t−2, d) weakly self-dual codes are evaluated, showing the upper bound is in fact attained, i.e., there are (14,6,6) and (38,18,12) weakly self-dual codes over GF(3).

An efficient means for finding the minimum weight of this family is given.

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References

  1. Wallis, W.D., Street, A.P., and Wallis, J.S., Combinatorics: Room Squares, sum free sets and Hadamard Matrices, in Lecture Notes in Mathematics, Vol. 292, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

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  2. Razen, R., Seberry, J., and Wehrhahn, K., Ordered partitions and codes generated by circulant matrices, J. Combinatorial Theory, Ser. A.

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  3. Mallows, C.L., Plass, V., and Sloane, N.J.H., Self-dual codes over GF(3), SIAM Journal of Applied Mathematics, Vol. 31, No. 4, 1976.

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  4. MacWilliams, F.J., and Sloane, N.J.A., The Theory of Error-Correcting Codes, North Holland, Amsterdam-New York-Oxford, 1977.

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© 1979 Springer-Verlag

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Rodger, C.A. (1979). A family of weakly self-dual codes. In: Horadam, A.F., Wallis, W.D. (eds) Combinatorial Mathematics VI. Lecture Notes in Mathematics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0102692

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  • DOI: https://doi.org/10.1007/BFb0102692

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09555-2

  • Online ISBN: 978-3-540-34857-3

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