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