engleski

New extremal Type II Z4-codes of length 32 obtained from Hadamard designs

For every Hadamard design with parameters 2−(n−1, n/2−1, n/4−1) having a skew-symmetric incidence matrix we give a construction of 54 Hadamard designs with parameters 2−(4n−1, 2n−1, n−1). This is a generalization of the construction given in[2]. For the case n=8 we construct doubly-even self-orthogonal binary linear codes from the corresponding Hadamard matrices of order 32. From these binary codes we construct five new extremal Type II Z4-codes of length 32, using the method given in [3]. The constructed codes are the first examples of extremal Type IIZ4-codes of length 32 and type 4^k1 2^k2, k1∈{;7, 8, 9, 10}; whose residue codes have minimum weight 8. Further, correcting the results from [1] we construct 5147 extremal Type II Z4-codes of length 32 and type 4^14 2^4. [1] K. H. Chan, Three New Methods for Construction of Extremal Type II Z4-Codes, PhDThesis (University of Illinois at Chicago, 2012) [2] D. Crnković, S. Rukavina, Some Symmetric (47, 23, 11) Designs, Glas. Mat. Ser. III, 38 (58)(2003) 1–9. [3] V. Pless, J. S. Leon, J. Fields, All Z4-codes of Type II and length 16 are known, J.Combin. Theory Ser. A, 78(1997) 32–50.

Hadamard design, Z4-code

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