316 research outputs found
A study of (xvt,xvt−1)-minihypers in PG(t,q)
AbstractWe study (xvt,xvt−1)-minihypers in PG(t,q), i.e. minihypers with the same parameters as a weighted sum of x hyperplanes. We characterize these minihypers as a nonnegative rational sum of hyperplanes and we use this characterization to extend and improve the main results of several papers which have appeared on the special case t=2. We establish a new link with coding theory and we use this link to construct several new infinite classes of (xvt,xvt−1)-minihypers in PG(t,q) that cannot be written as an integer sum of hyperplanes
Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18
All generalized Hadamard matrices of order 18 over a group of order 3,
H(6,3), are enumerated in two different ways: once, as class regular symmetric
(6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a
group of order 3 acting semi-regularly on points and blocks, and secondly, as
collections of full weight vectors in quaternary Hermitian self-dual codes of
length 18. The second enumeration is based on the classification of Hermitian
self-dual [18,9] codes over GF(4), completed in this paper. It is shown that up
to monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and
245 inequivalent Hermitian self-dual codes of length 18 over GF(4).Comment: 17 pages. Minor revisio
Families of explicitly isogenous Jacobians of variable-separated curves
We construct six infinite series of families of pairs of curves (X,Y) of
arbitrarily high genus, defined over number fields, together with an explicit
isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by
2, 3, or 4. For each family, we compute the isomorphism type of the isogeny
kernel and the dimension of the image of the family in the appropriate moduli
space. The families are derived from Cassou--Nogu\`es and Couveignes' explicit
classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is
reducible
Binary Hamming codes and Boolean designs
In this paper we consider a finite-dimensional vector space P over the Galois field GF(2), and the family Bk (respectively, B 17k) of all the k-sets of elements of P (respectively, of P 17=P 16{0}) summing up to zero. We compute the parameters of the 3-design (P,Bk) for any (necessarily even) k, and of the 2-design (P 17,B 17k) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of P, respectively of P 17, that induce permutations of Bk, respectively of B 17k. In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight
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