36,453 research outputs found

    Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

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    H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.Comment: 16 page

    A class of narrow-sense BCH codes over Fq\mathbb{F}_q of length qm−12\frac{q^m-1}{2}

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    BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. This paper studies a class of linear codes of length qm−12 \frac{q^m-1}{2} over Fq\mathbb{F}_q with special trace representation, where qq is an odd prime power. With the help of the inner distributions of some subsets of association schemes from bilinear forms associated with quadratic forms, we determine the weight enumerators of these codes. From determining some cyclotomic coset leaders δi\delta_i of cyclotomic cosets modulo qm−12 \frac{q^m-1}{2}, we prove that narrow-sense BCH codes of length qm−12 \frac{q^m-1}{2} with designed distance δi=qm−qm−12−1−q⌊m−32⌋+i−12\delta_i=\frac{q^m-q^{m-1}}{2}-1-\frac{q^{ \lfloor \frac{m-3}{2} \rfloor+i}-1}{2} have the corresponding trace representation, and have the minimal distance d=δid=\delta_i and the Bose distance dB=δid_B=\delta_i, where 1≤i≤⌊m+34⌋1\leq i\leq \lfloor \frac{m+3}{4} \rfloor
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