Abstract
Codes \({\mathcal{C}}(m,r)\) 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 \({\mathcal{C}}(m,r)\).
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Roth, R.M. Spectral-Null Codes and Null Spaces of Hadamard Submatrices. Designs, Codes and Cryptography 9, 177–191 (1996). https://doi.org/10.1023/A:1018018114369
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DOI: https://doi.org/10.1023/A:1018018114369