On nonsingularity of circulant matrices. (English) Zbl 1458.15049
Summary: In Communication theory and Coding, it is expected that certain circulant matrices having \(k\) ones and \(k+1\) zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when \(2k+1\) is either a power of a prime, or a product of two distinct primes. For any other integer \(2k+1\) we construct circulant matrices having determinant 0. The smallest singular matrix appears when \(2k+1=45\). The possibility for such matrices to be singular is rather low, smaller than \(10^{-4}\) in this case.
MSC:
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 11R18 | Cyclotomic extensions |
| 68P30 | Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science) |
| 94A05 | Communication theory |
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