2018/07/27 by Ronald Orozco López, López, Ronald Orozco · 1 citation
Computer Science · Engineering · Mathematics · #Algorithm #Arithmetic #Autocorrelation #Binary number #Block code #Circulant matrix #Coding theory and cryptography #Combinatorics #Combinatorics (math.CO) #Complex Hadamard matrix #Discrete mathematics #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Hadamard matrix #Hadamard transform #Hamming code #Mathematical analysis #Mathematics #Pseudorandom binary sequence #Statistics #graph theory and CDMA systems #math.CO #math.GR
paper · pdf · open access · doi:10.48550/arxiv.1807.10849
published in arXiv (Cornell University) (Cornell University) · 23 pages
arxiv created 2018/07/27 · openalex publication_date 2018/07/27 · arxiv updated 2018/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper three Schur ring are discussed, namenly: Hamming, circulant orbists and decimated circulant orbits Schur ring. By using autocorrelation function and the run structure of binary sequences we proof the relation between this Schur ring and combinatorial structures such as Hadamard matrices, periodic compatible binary sequences and perfect binary sequences. Cai proved for binary sequences that the autocorrelation function is in fact completely determined by its run structure. Also, he characterised the structure of the circulant Hadamard matrices. We characterise a more general structure, called periodic compatible binary sequences (PComS for brevety), which generalises Hadamard matrices, periodic complementary binary sequences and binary sequences with 2-level autocorrelation. Families of periodic compatibles binary sequences are presented. Also, we compute a bounds on familias PComS in Hamming Schur ring. The results obtained are applied to families of PComS such as circulant, with one and two circulant cores, Goethals-Seidel type and partial Hadamard matrices and perfect binary sequences.