On $c$-Hadamard Matrices

Abstract

We reintroduce the problem of finding square $\pm 1$-matrices, denoted $c\text{-}H(n)$, of order $n$, whose rows have non-zero inner product $c$. We obtain some necessary conditions for the existence of $c\text{-}H(n)$ and provide a characterization in terms of SBIBD parameters. Several new $c\text{-}H(n)$ constructions are given and new connections to Hadamard matrices and $D$-optimal designs are also explored.