Spectral-null codes and null spaces of Hadamard submatrices (Q679429)

A code-word \((c_0,\dots,c_{n-1})\) over \(\{-1,+1\}\subseteq\mathbb{R}\) is said to have an \(r\)th order spectral null at zero frequency if \(\sum_{j=0}^{n-1}j^i\cdot c_j=0\) for \(i=0,\dots,{n-1}\). The author considers codes \({\mathcal C}(m,r)\) of length \(2^m\) over \(\{-1,+1\}\) whose codewords have an \(r\)th order spectral null at zero frequency. The codes \({\mathcal C}(m,r)\) are defined as null spaces of truncations of Sylvester-type Hadamard matrices. It is shown that \({\mathcal C}(m,r)\) has minimum distance \(d_{\text{min}}=2^r\), and its redundancy has upper bound \(\text{red}({\mathcal C}(m,r))\leq\log_2(2^m+1)\cdot\sum_{i=0}^{r-1}{m\choose i}\). The proof of these properties uses the close connection between certain truncations of Hadamard matrices and parity-check matrices of binary Reed-Muller codes.