On One Construction Method for Hadamard Matrices

• Published in: Problems of information transmission Vol. 58; no. 4; pp. 306 - 328
• Main Authors: Villanueva, M., Zinoviev, V. A., Zinoviev, D. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-10-2022; Springer Nature B.V
• Subjects: 639/301/1034/1035; 639/925/357/995; Binary codes; Codes; Coding Theory; Communications Engineering; Construction; Control; Electrical Engineering; Engineering; Error correcting codes; Information Storage and Retrieval; Kernels; Networks; Systems Theory; Hadamard matrix; generalized concatenated construction; Hadamard code; Sylvester construction; rank of an Hadamard matrix; Kronecker product; nonequivalent Hadamard matrices; kernel dimension of an Hadamard matrix; code in the Lee metric
• ISSN: 0032-9460; 1608-3253
• DOI: 10.1134/S0032946022040032

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
].