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New Designs for Signal Sets with Low Cross-correlation, Balance Property and Large Linear Span: GF(p) Case
- IEEE Trans. Inform. Theory
, 2000
"... New designs for families of sequences over GF (p) with low cross correlation, balance property and large linear span are presented. The key idea of the new designs is to use short p-ary sequences of period v with the 2-level auto correlation function to construct a set of long sequences with the des ..."
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New designs for families of sequences over GF (p) with low cross correlation, balance property and large linear span are presented. The key idea of the new designs is to use short p-ary sequences of period v with the 2-level auto correlation function to construct a set of long sequences with the designed properties. The resulting sequences are of interleaved sequences of period v 2 . There are v cyclically shift distinct sequences in each family. The maximal magnitude of cross/out-of-phase auto correlation of sequences in the family is 2v + 3 which is optimal with respect to the Welch bound. In particular, for binary case, cross/out-of-phase auto correlation values belong to the set f1; v; v + 2; 2v + 3; 2v 1g. Each sequence in the family is balanced and has large linear span. For binary case, any sequence in such a family where the short sequences are quadratic residue sequences achieves the maximal linear span. For non-binary case, the new design gives the first type of signal set...
Character Sums and Polyphase Sequence Families with Low Correlation, DFT and Ambiguity
"... We present a survey on the current status of the constructions of polyphase sequences with low correlation, discrete Fourier transform (DFT), and ambiguity in both time and phase domain, including some new insights and results. Firstly, we systematically introduce the concepts of phase-shift operato ..."
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We present a survey on the current status of the constructions of polyphase sequences with low correlation, discrete Fourier transform (DFT), and ambiguity in both time and phase domain, including some new insights and results. Firstly, we systematically introduce the concepts of phase-shift operators and ambiguity functions of sequences, and give a new construction of polyphase sequences from combinations of different indexing field elements and hybrid characters. We then present the constructions, some known and some new, of polyphase sequences with low degree polynomials, for their low correlation, DFT and ambiguity can be bounded by directly applying the Weil bounds. Thirdly, we introduce the Hadamard equivalence, restate the conjectured new ternary 2-level autocorrelation sequences, and present their Hadamard equivalence relations. Some open problems are presented.
New Ternary and Quaternary Sequences with Two-Level Autocorrelation
"... Pseudorandom sequences with good correlation properties are widely used in communications and cryptography. The search of new sequences with two-level autocorrelation has been a very interesting problem for decades. In 2002, Gong and Golomb proposed the iterative decimation-Hadamard transform (DHT) ..."
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Pseudorandom sequences with good correlation properties are widely used in communications and cryptography. The search of new sequences with two-level autocorrelation has been a very interesting problem for decades. In 2002, Gong and Golomb proposed the iterative decimation-Hadamard transform (DHT) which is an useful tool to study two-level autocorrelation sequences. They showed that for all odd n ≤ 17, using the second-order decimation-Hadamard transform, and starting with a single binary m-sequence, all known two-level autocorrelation sequences of period 2 n − 1 which have no subfield factorization can be obtained. In this paper, we find many new ternary or quaternary sequences with two-level autocorrelation using the second-order decimation-Hadamard transform. The period of such sequences is 2 n − 1. Index Terms. Pseudorandom sequence, ternary sequence, quaternary sequence, two-level autocorrelation, iterative decimation-Hadamard transform (DHT), Dobbertin’s polynomial. 1
1 On a Connection between Ideal Two-level Autocorrelation and Almost Balancedness of p-ary Sequences*
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The Proof of Lin’s Conjecture via the Decimation-Hadamard Transform
"... In 1998, Lin presented a conjecture on a class of ternary sequences with ideal 2-level autocor-relation in his Ph.D thesis. Those sequences have a very simple structure, i.e., their trace repre-sentation has two trace monomial terms. In this paper, we present a proof for the conjecture. The mathemat ..."
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In 1998, Lin presented a conjecture on a class of ternary sequences with ideal 2-level autocor-relation in his Ph.D thesis. Those sequences have a very simple structure, i.e., their trace repre-sentation has two trace monomial terms. In this paper, we present a proof for the conjecture. The mathematical tools employed are the second-order multiplexing decimation-Hadamard transform, Stickelberger’s theorem, the Teichmüller character, and combinatorial techniques for enumerating the Hamming weights of ternary numbers. As a by-product, we also prove that the Lin conjectured ternary sequences are Hadamard equivalent to ternary m-sequences. Index Terms. Teichmüller character, decimation-Hadamard transform, multiplexing decimation-Hadamard transform, Stickelberger’s theorem, two-level autocorrelation. 1
GENERALIZED P-ARY SEQUENCES WITH TWO-LEVEL AUTOCORRELATION
"... In this paper, we find a family of-ary sequences with ideal two-level autocorrelation with symbols in the finite field. The proposed family may be considered as a generalization of the well-known nonbinary sequences introduced by Helleseth and Gong. Using the constructed sequences and-sequences, we ..."
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In this paper, we find a family of-ary sequences with ideal two-level autocorrelation with symbols in the finite field. The proposed family may be considered as a generalization of the well-known nonbinary sequences introduced by Helleseth and Gong. Using the constructed sequences and-sequences, we present a family of-ary sequences of which the correlation property is optimal in terms of the Welch lower bound.
