Binary Alexis sequences with perfect correlation

Binary Alexis (1986) sequences are useful signals for fast start-up equalization, channel estimation, synchronization, or ranging. The aperiodic autocorrelation functions of Alexis sequences vanish in a broad window region when correlated using additional binary preamble and postamble sequences. Originally, Alexis sequences were found by computer search. Two methods are presented by which such sequences can be constructed for any length N=p^a and N=2(p^a+1), where p, is an odd prime and a=1, 2, 3     

Linear complexity of de Bruijn sequences-old and new results

The linear complexity of a de Bruijn sequence is the degree of the shortest linear recursion which generates the sequence. It is well known that the complexity of a binary de Bruijn sequence of length 2ⁿ is bounded below by 2^n-1+n and above by 2^n-1 for n⩾3. We briefly survey the known knowledge in this area. Some new results are also presented, in particular, it is shown that for each interval of length 2^{[log n]+1} in the above range, there exist binary de Bruijn sequences of length 2ⁿ with linear complexity in the interval     

New designs for signal sets with low cross correlation, balanceproperty, and large linear span: GF(p) case

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 υ with the two-level autocorrelation function together with the interleaved structure to construct a set of long sequences with the desired properties. The resulting sequences are interleaved sequences of period υ². There are υ cyclically shift distinct sequences in each family. The maximal correlation value is 2υ + 3 which is optimal with respect to the Welch bound. Each sequence in the family is balanced and has large linear span. In particular, for binary case, cross/out-of-phase autocorrelation values belong to the set {1, -υ, υ + 2, 2υ + 3, -2υ - 1}, any sequence where the short sequences are quadratic residue sequences achieves the maximal linear span. It is shown that some families of these sequences can be implemented efficiently in both hardware and software     

Wide-sense and strict-sense nonblocking operation of multicastmulti-log2 n switching networks

Multicast connections are used in broad-band switching networks as well as in parallel processing. We consider wide-sense and strict-sense nonblocking conditions for multi-log₂ N switching networks with multicast connections. We prove that such networks are wide-sense nonblocking if they are designed by vertically stacking at least t · 2^n-t-1 + 2 ^n-2t-1 planes of a log₂ N networks together, where 1 ⩽ t ⩽ [n/2] and t defines the size of a blocking window K = 2^t. For t = [n/2] and n even, and for [n/2] ⩽ t ⩽ n the number of planes must be at least t · 2^n-t-1 + 1 and 2^t + (n - t - 1) · 2^n-t-1 - 2^2t-n-1 + 1, respectively. In the case of strict-sense nonblocking switching networks, the number of planes is at least N/2. The results obtained in this paper show that in many cases number of planes in wide-sense nonblocking switching networks is less than those for t = [n/2] considered by Tscha and Lee (see ibid., vol.47, p.1425-31, Sept. 1999). The number of planes given in the paper is the minimum number of planes needed for wide-sense nonblocking operation provided that Algorithm 1 is used for setting up connections. The minimum number of planes for such operation in general is still open issue     

Irreducible Polynomials Which Divide Trinomials Over GF (2)

The simplest linear shift registers to generate binary sequences involve only two taps, which corresponds to a trinomial over GF(2). It is therefore of interest to know which irreducible polynomials f(x) divide trinomials over GF(2), since the output sequences corresponding to f(x) can be obtained from a two-tap linear feedback shift register (with a suitable initial state) if and only if f(x) divides some trinomial t(x)=x^m+x^a+1 over GF(2). In this paper, we develop the theory of which irreducible polynomials do, or do not, divide trinomials over GF(2). Then some related problems such as Artin's conjecture about primitive roots, and the conjectures of Blake, Gao, and Lambert, as well as of Tromp, Zhang, and Zhao are discussed     

Generalized Kasami Sequences: The Large Set

In this correspondence, new binary sequence families F^k of period 2ⁿ-1 are constructed for even n and any k with gcd(k,n)=2 if n/2 is odd or gcd(k,n)=1 if n/2 is even. The distribution of their correlation values is completely determined. These families have maximum correlation 2^n/2+1 and family size 2^3n/2 + 2^n/2 for odd n/2 or 2^3n/2+2^n/2-1 for even n/2. The proposed families include the large set of Kasami sequences, where the k is taken as k=n/2+1.     

New construction for families of binary sequences with optimalcorrelation properties

We present a construction, in a closed form, for an optimal family of 2^m binary sequences of period 2^2m-1 with respect to Welch's (1974) bound, whenever there exists a balanced binary sequence of period 2^m-1 with ideal autocorrelation property using the trace function. This construction enables us to reinterpret a small set of Kasami and No (1988) sequences as a family constructed from m-sequences. New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period, Hall's sextic residue sequences, and miscellaneous sequences of unknown type. In addition, we enumerate the number of distinct families of binary sequences, which are constructed from a given binary sequence by this method     

Binary m-sequences with three-valued crosscorrelation: a proof ofWelch's conjecture

We prove the long-standing conjecture of Welch stating that for odd n=2m+1, the power function x^d with d=2^m+3 is maximally nonlinear on GF(2ⁿ) or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2^m+3 takes on precisely the three values -1, -1±2^m+1     

Large families of quaternary sequences with low correlation

A family of quaternary (Z₄-alphabet) sequences of length L=2^r-1, size M⩾L²+3L+2, and maximum nontrivial correlation parameter C_max⩽2√(L+1)+1 is presented. The sequence family always contains the four-phase family 𝒜. When r is odd, it includes the family of binary Gold sequences. The sequence family is easily generated using two shift registers, one binary, the other quaternary. The distribution of correlation values is provided. The construction can be extended to produce a chain of sequence families, with each family in the chain containing the preceding family. This gives the design flexibility with respect to the number of intermittent users that can be supported, in a code-division multiple-access cellular radio system. When r is odd, the sequence families in the chain correspond to shortened Z₄-linear versions of the Delsarte-Goethals codes     

Enumeration and criteria for cyclically shift-distinct GMWsequences

Gordon-Mills-Welch (GMW) sequences (also called cascaded GMW sequences) have two-level autocorrelations. This property makes them widely used in various communication and cryptographic systems. The generation of q-ary GMW sequences of period q^n-1 involves three types of parameters. To determine whether GMW sequences are cyclically shift-distinct for differing parameters has remained an open question until now. In this paper, we completely solve this problem for varying all three types of parameters. We find a criterion for cyclically shift-distinct q-ary GMW sequences of period q^n-1, and obtain the number of such sequences. For the special case of q=2, this solution facilitates counting the number of cyclic Hadamard difference sets which correspond to binary GMW sequences of period 2^n-1     

Asymptotic average redundancy of Huffman (and other) block codes

We study asymptotically the redundancy of Huffman (and other) codes. It has been known from the inception of the Huffman (1952) code that in the worst case its redundancy-defined as the excess of the code length over the optimal (ideal) code length-is not more than one. However, to the best of our knowledge no precise asymptotic results have been reported in literature thus far. We consider here a memoryless binary source generating a sequence of length n distributed as binomial (n, p) with p being the probability of emitting 0. Based on the results of Stubley (1994), we prove that for p<1/2 the average redundancy R¯_n^H of the Huffman code becomes as n→∞: R¯_n^H={(3/2-(1/ln2+o(1))=0.057304…, α irrational); (3/2-(1/M)(〈βMn〉-½)); (-(1/M(1-2^-1M/))2^-〈nβM〉M/); (+O(ρⁿ), α=N/M rational); where α=log₂ (1-p)/p and β=-log₂(1-p), ρ<1, M, N are integers such that gcd (N, M)=1, and 〈x〉=x-[x] is the fractional part of x. The appearance of the fractal-like function 〈βMn〉 explains the erratic behavior of the Huffman redundancy, and its “resistance” to succumb to a precise analysis. As a side result, we prove that the average redundancy of the Shannon block code is as n→∞: R¯_n^S{(½+o(1), α irrational); (½-1/M (〈Mnβ〉-½)); (+O(ρⁿ), α=N/M rational); where ρ<1. Finally, we derive the redundancy of the Golomb (1966) code (for the geometric distribution) which can be viewed as a special case of the Huffman and Shannon codes, Golomb's code redundancy has only oscillating behavior (i.e., there is not convergent mode). These findings are obtained by analytic methods such as theory of distribution of sequences modulo 1 and Fourier series     

Ternary m-sequences with three-valued cross-correlation function:new decimations of Welch and Niho type

We show that the cross correlation between two ternary m-sequences of period 3ⁿ-1 that differ by the decimation d=2·3^m+1, where n=2m+1, takes on three different values. We conjecture the same result for the decimation d=2·3 ^r+1, where n is odd and r is defined by the condition 4r+1≡0 mod n. These two new cases form in a sense ternary counterparts of two previously confirmed binary cases, the conjectures of Welch and Niho (1972)     

A new family of binary pseudorandom sequences having optimalperiodic correlation properties and larger linear span

A collection of families of binary {0,1} pseudorandom sequences is introduced. Each sequence within a family has period N=2"-1, where n=2m is an even integer. There are 2^m sequences within a family, and the maximum overall (nontrivial) auto- and cross-correlation values equals 2^m+1. Thus, these sequences are optimal with respect to the Welch bound on the maximum correlation value. Each family contains a Gordon-Mills-Welch (GMW) sequence, and the collection of families includes as a special case the small set of Kasami sequences. The linear span of these sequences varies within a family but is always greater than or equal to the linear span of the GMW sequence contained within the family. Exact closed-form expressions for the linear span of each sequence are given. The balance properties of such families are evaluated, and a count of the number of distinct families of given period N that can be constructed is provided     

Fast conversion between binary and residue numbers

New, efficient hardware implementations are considered which perform code conversions between the three moduli (2ⁿ-1, 2 ⁿ, 2ⁿ+1) residue number systems and their binary representations. Significant hardware saving together with high-speed throughput is achieved by using only n and (n+1)-bit binary adders     

Family of uncorrelated binary ZCZ sequence pairs with mismatched filtering

A family of uncorrelated binary sequence pairs of length N=4(p^m+1) with zero correlation zone (ZCZ) (N/4) -1 and mismatched filtering is proposed. The sequence pairs are derived from almost- perfect binary and ternary sequences and possess energy efficiency close to 1 as N becomes large. They can be used in quasi-synchronous code division multiple access and radar systems to reduce interference.     

Sequences with almost perfect linear complexity profiles and curvesover finite fields

For stream ciphers, we need to generate pseudorandom sequences which are of properties of unpredictability and randomness. A important measure of unpredictability and randomness is the linear complexity profile (l.c.p.) l_a(n) of a sequence a. A sequence a is called almost perfect if the l.c.p. is l_a(n)=n/2+O(1). Based on curves over finite fields, we present a method to construct almost perfect sequences. We also illustrate our construction by explicit examples from the projective line and elliptic curves over the binary field     

RNS-To-Binary Converter for a New Three-Moduli Set {2n+1−1,2n,2n−1}

In this brief, the design of residue number system (RNS) to binary converters for a new powers-of-two related three-moduli set {2ⁿ⁺¹ - 1, 2ⁿ, 2ⁿ - 1} is considered. This moduli set uses moduli of uniform word length (n to n + 1 bits). It is derived from a previously investigated four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2^{n +1} - 1}. Three RNS-to-binary converters are proposed for this moduli set: one using mixed radix conversion and the other two using Chinese remainder theorem. Detailed architectures of the three converters as well as comparison with some earlier proposed converters for three-moduli sets with uniform word length and the four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ⁺¹ - 1} are presented.     

4-phase sequences with near-optimum correlation properties

Two families of four-phase sequences are constructed using irreducible polynomials over Z₄. Family A has period L =2^r-1. size L+2. and maximum nontrivial correlation magnitude C_max⩽1+√(L+1), where r is a positive integer. Family B has period L=2(2^r-1). size (L+2)/4. and C_max for complex-valued sequences. Of particular interest, family A has the same size and period as the family of binary Gold sequences. but its maximum nontrivial correlation is smaller by a factor of √2. Since the Gold family for r odd is optimal with respect to the Welch bound restricted to binary sequences, family A is thus superior to the best possible binary design of the same family size. Unlike the Gold design, families A and B are asymptotically optimal whether r is odd or even. Both families are suitable for achieving code-division multiple-access and are easily, implemented using shift registers. The exact distribution of correlation values is given for both families     

The weights of the orthogonals of the extended quadratic binaryGoppa codes

Starting from results on elliptic curves and Kloosterman sums over the finite field GE(2^t), the authors determine the weights of the orthogonals of some binary linear codes; the Melas code of length, the irreducible cyclic binary code of length 2^t+1, and the extended binary Goppa codes defined by polynomials of degree two     

Low-resistance bandgap-engineeredW/Si1-xGex/Si contacts

Bandgap-engineered W/Si_1-xGe_x/Si junctions (p⁺ and n⁺) with ultra-low contact resistivity and low leakage have been fabricated and characterized. The junctions are formed via outdiffusion from a selectively deposited Si_0.7Ge _0.3 layer which is implanted and annealed using RTA. The Si _1-xGe_x layer can then be selectively thinned using NH₄OH/H₂O₂/H₂O at 75°C with little change in characteristics or left as-deposited. Leakage currents were better than 1.6×10^-9 A/cm² (areal), 7.45×10^-12 A/cm (peripheral) for p⁺/n and 3.5×10^-10 A/cm² (peripheral) for n⁺/p. W contacts were formed using selective LPCVD on Si_1-xGe_x. A specific contact resistivity of better than 3.2×10^-8 Ω cm² for p ⁺/n and 2.2×10^-8 Ω cm² for n⁺/p is demonstrated-an order of magnitude n⁺ better than current TiSi₂ technology. W/Si_1-xGe _x/Si junctions show great potential for ULSI applications