Spectral-Null Codes and Null Spaces of Hadamard Submatrices

AbstractCodes $${\mathcal{C}}(m,r)$$ of length 2m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of $${\mathcal{C}}(m,r)$$ all have an rth order spectral null at zero frequency. Establishing the connection between $${\mathcal{C}}(m,r)$$ and the parity-check matrix of Reed-Muller codes, the minimum distance of $${\mathcal{C}}(m,r)$$ is obtained along with upper bounds on the redundancy of $${\mathcal{C}}(m,r)$$ . An efficient algorithm is presented for encoding unconstrained binary sequences into $${\mathcal{C}}(m,r)$$ .