Hadamard matrices and doubly even self-dual error-correcting codes

AbstractLet n be an integer with n≡4 (mod 8). For any Hadamard matrices Hn of order n, we give a method to define a doubly even self-dual [2n, n] code C(NHn). Then we will prove that two Hadamard equivalent matrices define equivalent codes

On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

AbstractWe construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason–MacWilliams group. We find this canonical set in the vector space (⊗i=1lV)G, where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants C [ x, y ]G0of the two-dimensional group G0of order 48 which is the intersection of G and SL(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group Sl. That is, by letting the symmetric group Slacts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group Slacts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (⊗i=1lV)G. This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms

Distinguishing Siegel theta series of degree 4 for the 32-dimensional even unimodular extremal lattices

In a previous paper we showed that if one particular Fourier coefficient of the Siegel theta series of degree 4 for a 32-dimensional even unimodular extremal lattice is known then the other Fourier coefficients of the series are in principle determined. In this paper we choose the quaternary positive definite symmetric matrix (Formula presented.), and calculate the Fourier coefficient (Formula presented.) of the Siegel theta series of degree 4 associated with the five even unimodular extremal lattices which come from the five binary self-dual extremal [32,16,8] codes. As a result we can show that the five Siegel theta series of degree 4 associated with the five 32-dimensional even unimodular extremal lattices are distinct. © 2016 Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin HeidelbergEmbargo Period 12 month