On the Classification of Cyclic Hadamard Sequences

Solomon W. GOLOMB

doi:10.1093/ietfec/e89-a.9.2247

On the Classification of Cyclic Hadamard Sequences

Solomon W. GOLOMB

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Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (v, k, λ)-designs. When v = 4t-1, k = 2t -1 and λ = t-1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, v is either a prime number, a product of twin primes, or one less than a power of 2. Except when v = 2^k-1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when v = p(p+2) where both p and p+2 are prime); or sextic residues (when v is a prime of the form 4a² + 27). However, when v = 2^k-1, many constructions are now known, including m-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2^k-1 is prime), GMW sequences and their generalizations (when k is composite), certain term-by-term sums of three and of five m-sequences and more general sums of trace terms, several constructions based on hyper-ovals in finite geometries (found by Segre, by Glynn, and by Maschietti), and the result of performing the Welch-Gong transformation on some of the foregoing.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E89-A No.9 pp.2247-2253

Publication Date: 2006/09/01

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Online ISSN: 1745-1337

DOI: 10.1093/ietfec/e89-a.9.2247

Type of Manuscript: Special Section INVITED PAPER (Special Section on Sequence Design and its Application in Communications)

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Solomon W. GOLOMB, "On the Classification of Cyclic Hadamard Sequences" in IEICE TRANSACTIONS on Fundamentals, vol. E89-A, no. 9, pp. 2247-2253, September 2006, doi: 10.1093/ietfec/e89-a.9.2247.
Abstract: Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (v, k, λ)-designs. When v = 4t-1, k = 2t -1 and λ = t-1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, v is either a prime number, a product of twin primes, or one less than a power of 2. Except when v = 2^k-1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when v = p(p+2) where both p and p+2 are prime); or sextic residues (when v is a prime of the form 4a² + 27). However, when v = 2^k-1, many constructions are now known, including m-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2^k-1 is prime), GMW sequences and their generalizations (when k is composite), certain term-by-term sums of three and of five m-sequences and more general sums of trace terms, several constructions based on hyper-ovals in finite geometries (found by Segre, by Glynn, and by Maschietti), and the result of performing the Welch-Gong transformation on some of the foregoing.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e89-a.9.2247/_p

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@ARTICLE{e89-a_9_2247,
author={Solomon W. GOLOMB, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={On the Classification of Cyclic Hadamard Sequences},
year={2006},
volume={E89-A},
number={9},
pages={2247-2253},
abstract={Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (v, k, λ)-designs. When v = 4t-1, k = 2t -1 and λ = t-1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, v is either a prime number, a product of twin primes, or one less than a power of 2. Except when v = 2^k-1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when v = p(p+2) where both p and p+2 are prime); or sextic residues (when v is a prime of the form 4a² + 27). However, when v = 2^k-1, many constructions are now known, including m-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2^k-1 is prime), GMW sequences and their generalizations (when k is composite), certain term-by-term sums of three and of five m-sequences and more general sums of trace terms, several constructions based on hyper-ovals in finite geometries (found by Segre, by Glynn, and by Maschietti), and the result of performing the Welch-Gong transformation on some of the foregoing.},
keywords={},
doi={10.1093/ietfec/e89-a.9.2247},
ISSN={1745-1337},
month={September},}

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TY - JOUR
TI - On the Classification of Cyclic Hadamard Sequences
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2247
EP - 2253
AU - Solomon W. GOLOMB
PY - 2006
DO - 10.1093/ietfec/e89-a.9.2247
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E89-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2006
AB - Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (v, k, λ)-designs. When v = 4t-1, k = 2t -1 and λ = t-1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, v is either a prime number, a product of twin primes, or one less than a power of 2. Except when v = 2^k-1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when v = p(p+2) where both p and p+2 are prime); or sextic residues (when v is a prime of the form 4a² + 27). However, when v = 2^k-1, many constructions are now known, including m-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2^k-1 is prime), GMW sequences and their generalizations (when k is composite), certain term-by-term sums of three and of five m-sequences and more general sums of trace terms, several constructions based on hyper-ovals in finite geometries (found by Segre, by Glynn, and by Maschietti), and the result of performing the Welch-Gong transformation on some of the foregoing.
ER -