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.
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