On One Construction Method for Hadamard Matrices.

Using a concatenated construction for -ary codes, we construct codes over in the Lee metrics which after a proper mapping to the binary alphabet (which in the case of is the well-known Gray map) become binary Hadamard codes (in particular, Hadamard matrices). Our construction allows to increase the rank and the kernel dimension of the resulting Hadamard code. Using computer search, we construct new nonequivalent Hadamard matrices of orders , , and with various fixed values of the rank and the kernel dimension in the range of possible values. It was found that in a special case, our construction coincides with the Kronecker (or Sylvester) construction and can be regarded as a version of a presently known [1] modified Sylvester construction which uses one Hadamard matrix of order and (not necessarily distinct) Hadamard matrices of order . We generalize this modified construction by proposing a more general Sylvester-type construction based on two families of (not necessarily distinct) Hadamard matrices, namely, on matrices of order and matrices of order . The resulting matrix is of order , as in the construction from [1]. [ABSTRACT FROM AUTHOR]