Nam l Yu

Constructions of quadratic bent functions in polynomial forms

In this correspondence, the constructions and enumerations of all bent functions represented by a polynomial form of f(x)=/spl Sigma//sub i=1//sup n/2-1/c/sub i/Tr(x/sup 1+2(i)/)+c/sub n/2/Tr/sub 1//sup n/ /sup /2/(x/sup 1+2(n/2)/), c/sub i//spl isin//sub 2/ F/sub 2/ are presented for special cases of n. Using an iterative approach, the construction of bent functions of n variables with degree n/2 is also provided using the constructed quadratic bent functions.

A new binary sequence family with low correlation and large size

For odd n=2l+1 and an integer /spl rho/ with 1/spl les//spl rho//spl les/l, a new family S/sub o/(/spl rho/) of binary sequences of period 2/sup n/-1 is constructed. For a given /spl rho/, S/sub o/(/spl rho/) has maximum correlation 1+2/sup n+2/spl rho/-1/2/, family size 2/sup n/spl rho//, and maximum linear span n(n+1)/2. Similarly, a new family of S/sub e/(/spl rho/) of binary sequences of period 2/sup n/-1 is also presented for even n=2l and an integer /spl rho/ with 1/spl les//spl rho/<l, where maximum correlation, family size, and maximum linear span are 1+2/sup n/2+/spl rho//,2/sup n/spl rho//, and n(n+1)/2, respectively. The new family S/sub o/(/spl rho/) (or S/sub e/(/spl rho/)) contains Boztas and Kumars construction (or Udayas) as a subset if m-sequences are excluded from both constructions. As a good candidate with low correlation and large family size, the family S/sub o/(2) is discussed in detail by analyzing its distribution of correlation values.

New Binary Sequences With Optimal Autocorrelation Magnitude

New binary sequences of period N = 4(2m - 1) for even m ges 4 are found, where the sequences are described by a 4 X (2m - 1) array structure. The new sequences are almost balanced and have four- valued autocorrelation, i.e., {N, 0, plusmn4}, which is optimal with respect to autocorrelation magnitude. The complete autocorrelation distribution and the exact linear complexity of the sequences are mathematically derived. Finally, it is shown that the sequences are implemented by a combination of linear feedback shift registers and a simple logic.

New Construction of

For prime <i>p</i> and a positive integer <i>m</i> , it is shown that <i>M</i>-ary Sidelnikov sequences of period <i>p</i>^2m-1, if <i>M</i> | <i>p</i>^m-1, can be equivalently generated by the operation of elements in a finite field <i>GF(p</i>^m), including a <i>p</i>^m-ary <i>m</i> -sequence. From the <i>(p</i>^m-1) ×(<i>p</i>^m+1) array structure of the sequences, it is then found that a half of the column sequences and their constant multiples have low correlation enough to construct new <i>M</i> -ary sequence families of period <i>p</i>^m-1. In particular, new <i>M</i>-ary sequence families of period <i>p</i>^m-1 are constructed from the combination of the column sequence families and known Sidelnikov-based sequence families, where the new families have larger family sizes than the known ones with the same maximum correlation magnitudes. Finally, it is shown that the new <i>M</i> -ary sequence family of period <i>p</i>^m-1 and the maximum correlation magnitude <i>2√{p</i>^m}+6 asymptotically achieves <i>√2</i> times the equality of the Sidelnikovs lower bound when <i>M</i>=<i>p</i>^m-1 for odd prime <i>p</i>.

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Power residue and Sidelnikov sequences are polyphase sequences with low correlation and variable alphabet sizes, represented by multiplicative characters. In this paper, sequence families constructed from the shift and addition of the polyphase sequences are revisited. Initially, ψ(0)=1 is assumed for multiplicative characters ψ to represent power residue and Sidelnikov sequences in a simple form. The Weil bound on multiplicative character sums is refined for the assumption, where the character sums are equivalent to the correlations of sequences represented by multiplicative characters. General constructions of polyphase sequence families that produce some of known families as the special cases are then presented. The refined Weil bound enables the efficient proofs on the maximum correlation magnitudes of the sequence families. From the constructions, it is shown that M-ary known sequence families with large size can be partitioned into (M+1) disjoint subsequence families with smaller maximum correlation magnitudes. More generalized constructions are also considered by the addition of multiple cyclic shifts of power residue and Sidelnikov sequences.

-Ary Sequence Families With Low Correlation From the Structure of Sidelnikov Sequences

An (N, K) codebook is a set of N unit-norm code vectors in a K-dimensional vector space. For its applications, it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible, meeting the Welch bound equality strictly or asymptotically. In this paper, an (N, K) codebook is constructed from a K × N partial matrix with K <; N, where each code vector is equivalent to a column of the matrix. To obtain the K × N matrix, K rows are selected from a J × N matrix Φ, associated with a binary sequence of length J and Hamming weight K, where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence. It is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of Φ-transform of the binary sequence. Thus, constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its Φ-transform is as small as possible. From the discovery, new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from Fourier and Hadamard matrices associated with binary sequences.

Multiplicative Characters, the Weil Bound, and Polyphase Sequence Families With Low Correlation

In this paper, a new class of Fourier-based matrices is studied for deterministic compressed sensing. Initially, a basic partial Fourier matrix is introduced by choosing the rows deterministically from the inverse discrete Fourier transform (DFT) matrix. By row/column rearrangement, the matrix is represented as a concatenation of DFT-based submatrices. Then, a full or a part of columns of the concatenated matrix is selected to build a new M × N deterministic compressed sensing matrix, where M = pr and N = L(M + 1) for prime p, and positive integers r and L ≤ M - 1. Theoretically, the sensing matrix forms a tight frame with small coherence. Moreover, the matrix theoretically guarantees unique recovery of sparse signals with uniformly distributed supports. From the structure of the sensing matrix, the fast Fourier transform (FFT) technique can be applied for efficient signal measurement and reconstruction. Experimental results demonstrate that the new deterministic sensing matrix shows empirically reliable recovery performance of sparse signals by the CoSaMP algorithm.

A Construction of Codebooks Associated With Binary Sequences

New families of near-complementary sequences are presented for peak power control in multicarrier communications. A framework for near-complementary sequences is given by an explicit Boolean expression and an equivalent matrix structure. The framework transforms seed pairs to near-complementary sequences by the aid of Golay complementary sequences. New families of near-complementary sequences of various lengths and PMEPR <; 4 are then presented, where the sequences are constructed by the framework employing the seeds of shortened and extended Golay complementary pairs. The families present in a constructive way a large number of sequences of PMEPR <; 4 for the lengths (<; 100) of 24, 28, 30, 34, 36, 48, 56, 60, 62, 66, 68, 72, and 96 where no Golay pairs have been reported. The sequence families can find potential applications for peak power control requiring codewords or sequences of various lengths as well as low peak-to-mean envelope power ratio (PMEPR).

Deterministic construction of Fourier-based compressed sensing matrices using an almost difference set

Compressed sensing is a novel technique where one can recover sparse signals from the undersampled measurements. In this paper, a K × N measurement matrix for compressed sensing is deterministically constructed via additive character sequences. The Weil bound is then used to show that the matrix has asymptotically optimal coherence for N = K2, and that it is a tight frame. A sparse recovery guarantee for the incoherent tight frame is also discussed. Numerical results show that the deterministic sensing matrix guarantees empirically reliable recovery performance via an l1-minimization method for noiseless measurements.

Near-Complementary Sequences With Low PMEPR for Peak Power Control in Multicarrier Communications

In this letter, a new class of real-valued matrices is presented for deterministic compressed sensing. A base matrix is constructed by cyclic shifts of binary sequences in an optical orthogonal code (OOC). Then, a Hadamard matrix is used for its extension, which ultimately produces a real-valued matrix that takes the entries of 0, -1 and +1 before normalization. The new sensing matrix forms a tight frame with small coherence, which theoretically guarantees the average recovery performance of sparse signals with uniformly distributed supports. Several example sensing matrices are presented by employing a special type of OOCs obtained from modular Golomb rulers.