Spectral-Null Codes and Null Spaces of Hadamard Submatrices

Authors:
Ron M. Roth
Affiliations:
Computer Science Department, Technion—/Israel Institute of Technology, Haifa 32000, Israel/ ronny@cs.technion.ac.il
Venue:
Designs, Codes and Cryptography
Year:
1996

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Abstract

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