generalized chirp like polyphase sequences with optimum correlation properties a new general class of polyphase sequences with ideal periodic autocorrelation function is presented the new class of sequences is based on the application of zadoff chu polyphase sequences of length n sm sup 2 where s and m are any positive integers it is shown that the generalized chirp like sequences of odd length have the optimum crosscorrelation function under certain conditions finally recently proposed generalized p4 codes are derived as a special case of the generalized chirp like sequence

free boundary regularity for a problem arising in superconductivity this paper concerns regularity properties of the mean field theory of superconductivity the problem is reminiscent of the one studied earlier by l a caffarelli l karp and h shahgholian in connection with potential theory the difficulty introduced in this paper is the existence of several patches where on each patch the solution to the problem may have different constant values however using a refined analysis we reduce the problem to the one patch case at least locally near regular free boundary points using a monotonicity formula due to georg s weiss we characterize global solutions of a related equation hence earlier regularity results apply and we conclude the c1 regularity of the free boundary

theory and applications of q ary interleaved sequences a new class of q ary sequences called interleaved sequence is introduced their periods shift equivalence linear spans and autocorrelation functions are derived the interleaved sequences include a large number of popular sequences such as multiplexed sequences clock controlled sequences kasami 1966 sequences gmw sequences geometric sequences and no 1989 sequences a special class of the interleaved sequences is constructed by mapping gf q sup m sequences into gf q sequences in terms of different bases of gf q sup m over gf q as an application of the theory of interleaved sequences some new families of binary pseudo random sequences are constructed which have large linear spans optimal periodic cross autocorrelation functions balance and the rapidly hopped properties a complete comparison of the new family of sequences with the gold sequence family the kasami small and large set sequence families the bent function sequence family and the no sequence family is discussed this shows that the new sequence families have important advantages for use in spread spectrum multiple access communication systems

classification error analysis in stereo vision abstract depth perception in humans is obtained by comparing images generated by the two eyes to each other given the highly stochastic nature of neurons in the brain this comparison requires maximizing the mutual information mi between the neuronal responses in the two eyes by distributing the coding information across a large number of neurons unfortunately mi is not an extensive quantity making it very difficult to predict how the accuracy of depth perception will vary with the number of neurons n in each eye to address this question we present a two arm distributed decentralized sensors detection model we demonstrate how the system can extract depth information from a pair of discrete valued stimuli represented here by a pair of random dot matrix stereograms using the theory of large deviations we calculated the rate at which the global average error probability of our detector and the mi between the two arms output vary with n we found that mi saturates exponentially with n at a rate which decays as 1 n the rate function approached the chernoff distance between the two probability distributions asymptotically our results may have implications in computer stereo vision that uses hebbian based algorithms for terrestrial navigation

bat chirps with good properties zak space construction of perfect polyphase sequences previously a discretization of the linear fm chirp of length n kl 2 l and kl isin z was given and the conditions for its minimal zak space support were derived chirps satisfying these conditions are known as finite chirps in this work subsets of finite chirps of length n l 2 l a prime are examined the investigation leads to a new zak space construction of general polyphase sequence sets of size l 1 with optimal auto and cross correlation properties known as perfect sequence sets it is shown that perfect sequence sets are closely related to sets of finite chirps and in particular include the sets of zadoff chu sequences which are identical with subsets of finite chirps and the sets of generalized frank sequences which are identical with sets of modulations of finite chirps as special cases the entire collection of perfect sequence sets is then given by a partition of the set of perfect auto correlation sequences obtained by right coset decomposition of the group of all permutations with respect to a certain cyclic group the construction suggests several further generalizations that can be obtained by operating exclusively on subgroups of the permutation group

the status of costas arrays the definition the basic properties and all the currently known systematic constructions for costas arrays are presented as well as a table of the number c n of costas arrays of order n for 2 les n les 26 it is proved that lim supnrarrinfin c n infin and the conjecture liminfnrarrinfin c n 0 is discussed a costas array of order n is known to be equivalent to a permutation 1 2 n for which the difference triangle contains no repeated elements in any row a generalized costas array of order n q 1 or n q 2 is defined as a permutation of the nonzero elements or also excluding 1 of the q element field for which the difference triangle contains no repeated elements in any row two new constructions for these generalized costas arrays are described and illustrated

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 kumar s construction or udaya s 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

sequence families with low correlation derived from multiplicative and additive characters for integer r satisfying 0 r p 2 a sequence family r of polyphase sequences of prime period p size p 2 pr and maximum correlation at most 2 r 1 p is presented the sequence families are nested that is r is contained in r 1 which provides design flexibility with respect to family size and maximum correlation the sequences in r are derived from a combination of multiplicative and additive characters of a prime field estimates on hybrid character sums are then used to bound the maximum correlation this construction generalizes 0 which was previously proposed by scholtz and welch sequence family 2 is closely related to a recent design by wang and gong who bounded its maximum correlation using methods from representation theory and asked for a more direct proof of this bound such a proof is given here and an improvement of the bound is provided

constructions of quasi complementary sequence sets associated with characters an boldsymbol m boldsymbol k boldsymbol n delta max quasi complementary sequence set qcss is referred to as a set of boldsymbol m 2 d matrices of order boldsymbol k times boldsymbol n with periodic tolerance delta max in a multicarrier code division multiple access mc cdma communication system the set size boldsymbol m of a qcss is equal to the maximum number of users it can support and the periodic tolerance delta max determines the interference performance for the application of a qcss it is desirable that the set size should be as large as possible and the periodic tolerance should be as small as possible in this paper a framework of a periodic qcss is proposed from additive and multiplicative characters associated with a specific integer set it is then discovered that the parameters of the obtained qcss are determined by the employed integer set from this discovery new classes of periodic qcsss with large set sizes and low periodic tolerances are constructed and associated with some known integer sets

multiplicative characters the weil bound and polyphase sequence families with low correlation 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 0 1 is assumed for multiplicative characters 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

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

modulatable orthogonal sequences and their application to ssma systems a set of n 1 orthogonal sequences of period n sup 2 is proposed where n is a natural number each orthogonal sequence proposed can be modulated by n complex numbers of absolute value 1 so the modulated sequence is also orthogonal when n is an odd prime number the absolute value of the cross correlation function between any two of the n 1 orthogonal sequences is constant and satisfies the mathematical lower bound this property of the cross correlation function is not changed when each of the two orthogonal sequences is modulated by n complex numbers of absolute value 1 two spread spectrum multiple access ssma systems using these sequences are proposed one system is an asynchronous ssma system using the proposed sequences unmodulated the cochannel interference peak between any two channels in this system realizes the mathematical lower bound for an asynchronous ssma system using a set of orthogonal sequences the other system is a synchronous ssma system without cochannel interference which uses the modulated form of the proposed sequences

the finite harmonic oscillator and its applications to sequences communication and radar a novel system called the oscillator system consisting of order of p3 functions signals on the finite field fp with p an odd prime is described and studied the new functions are proved to satisfy good autocorrelation cross correlation and low peak to average power ratio properties moreover the oscillator system is closed under the operation of discrete fourier transform applications of the oscillator system for discrete radar and digital communication theory are explained finally an explicit algorithm to construct the oscillator system is presented

prime phase sequences with periodic correlation properties better than binary sequences for the case where p is an odd prime n or 2 is an integer and omega is a complex primitive pth root of unity a construction is presented for a family of p sup n p phase sequences symbols of the form omega sup i where each sequence has length p sup n 1 and where the maximum nontrivial correlation value c sub max does not exceed 1 square root p sup n a complete distribution of correlation values is provided as a special case of this construction a previous construction due to sidelnikov 1971 is obtained the family of sequences is asymptotically optimum with respect to its correlation properties and in comparison with many previous nonbinary designs the present design has the additional advantage of not requiring an alphabet of size larger than three the new sequences are suitable for achieving code division multiple access and are easily implemented using shift registers they wee discovered through an application of deligne s bound 1974 on exponential sums of the weil type in several variables the sequences are also shown to have strong identification with certain bent functions

binary sequence sets with favorable correlations from difference sets and mds codes we propose new families of pseudorandom binary sequences based on hadamard difference sets and mds codes we obtain for p 4k 1 prime and t an integer with 1 spl les t spl les p 1 2 a set of p sup t binary sequences of period p sup 2 whose peak correlation is bounded by 1 2t p 1 the sequences are balanced have high linear complexity and are easily generated

regularity of a free boundary in parabolic potential theory we study the regularity of the free boundary in a stefan type problem 1u t u in d rn r u u 0 on d with no sign assumptions on u and the time derivative t u

a note on complex sequences with low correlations corresp a construction technique is given for complex sequences each component of magnitude unity with iow autocorrelation and cross correlation the construction is based on an application of the carlitz uchiyama theorem to cyclic equivalence classes of maximal period of reed solomon codes over gf p p being an odd prime

n shift cross orthogonal sequences n shift cross orthogonal sequences and a complete complementary code derived from them are defined and discussed a general method for generating this code is also discussed a synchronous multiple user spread spectrum multiple system is proposed that uses n shift cross orthogonal sequences

new sequences design from weil representation with low two dimensional correlation in both time and phase shifts a new elementary expression of the construction first proposed by gurevich hadani and sochen is given which avoids the explicit use of the weil representation the sequences in this signal set are given by both multiplicative character and additive character of finite field fp such a signal set consists of p2 p 2 time shift distinct sequences the magnitude of the two dimensional autocorrelation function i e the ambiguity function in both time and phase of each sequence is upper bounded by 2 p at any shift not equal to 0 0 furthermore the magnitude of their fourier transform spectrum is less than or equal to 2 for a subset consisting of p p 2 phase shift distinct sequences in this signal set the magnitude of the ambiguity function of any pair is upper bounded by 4 p a proof is given through finding a new expression of the sequences in the finite harmonic oscillator system an open problem for directly establishing these assertions without involving the weil representation is addressed

new polyphase sequence families with low correlation derived from the weil bound of exponential sums in this paper the sequence families of which maximum correlation is determined by the weil bound of exponential sums are revisited using the same approach two new constructions with large family sizes and low maximum correlation are given the first construction is an analog of one recent result derived from the interleaved structure of sidel nikov sequences for a prime p and an integer m p 1 the new m ary sequence families of period p are obtained from irreducible quadratic polynomials and known power residue based sequence families the new sequence families increase family sizes of the known power residue based sequence families but keep the maximum correlation unchanged in the second construction the sequences derived from the weil representation are generalized where each new sequence is the elementwise product of a modulated sidel nikov sequence and a modulated trace sequence for positive integers d p and m pn 1 the new family consists of m 1 pnd sequences with period pn 1 alphabet size mp and the maximum correlation bounded by d 1 pn 3

geometric and energetic criteria for the free boundary regularity in an obstacle type problem we consider an obstacle type problem u f x in d u u 0 on d where d is a given open set in n and is an unknown open subset of d the problem originates in potential theory in connection with harmonic continuation of potentials the qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign using geometric and energetic criteria in delicate combination we show the c1 1 regularity of the solutions and the regularity of the free boundary below the lipschitz threshold for the right hand side

delay doppler channel estimation in almost linear complexity a fundamental task in wireless communication is channel estimation compute the channel parameters a signal undergoes while traveling from a transmitter to a receiver in the case of delay doppler channel i e a signal undergoes only delay and doppler shifts a widely used method to compute the delay doppler parameters is the matched filter algorithm it uses a pseudo random sequence of length n and in case of non trivial relative velocity between transmitter and receiver its computational complexity is o n2logn in this paper we introduce a novel approach of designing sequences that allow faster channel estimation using group representation techniques we construct sequences which enable us to introduce a new algorithm called the flag method that significantly improves the matched filter algorithm the flag method finds m delay doppler parameters in o mnlogn operations we discuss applications of the flag method to gps and radar systems

some new monotonicity theorems with applications to free boundary problems the monotonicity theorem of alt caffarelli and friedman acf lemma 5 1 plays a central role in the existence and regularity theory of two phase free boundary problems it has been extended for example to variable coefficient operators c3 lemma 1 and to eigenvalue problems fl in this paper we reformulate the theorem so that it applies to inhomogeneous equations in which the right hand side of the equation need not vanish at the free boundary of particular interest is the case in which au takes two different constant values on the set where u 0 and on the set where u 0 the prandtl batchelor