Self-avoiding random loops

A random loop, or polygon, is a simple random walk whose trajectory is a simple Jordan curve. The study of random loops is extended in two ways. First, the probability P_n(x,y) that a random n-step loop contains a point (x,y) in the interior of the loop is studied, and (1/2, 1/2) is shown to be (1/2)-(1/ n). It is plausible that P_n(x,y) tends toward 1/2 for all ( x,y), but this is not proved even for (x,y)=(3/2,1/2) A way is offered to simulate random n-step self-avoiding loops. Numerical evidence obtained with this simulation procedure suggests that the probability P_n (3/2,1/2)≈(1/2)-(c/n), for some fixed c      

A generalization of consecutive-k-out-of-n:Fsystems

A linear (m, n)-lattice system consists of m ·n elements arranged like the elements of a (m ,n)-matrix, i.e. each of the m rows includes m elements, and each of the n columns includes m elements. A circular (m,n)-lattice system consists of m circles (centered at the same point) and n rays. The intersections of the circle and the rays represent the elements, i.e. each of the circles includes n elements and each of the rays has m elements. A (linear or circular) (m, n)-lattice system is a (linear or circular) connected-X-out-of-(m,n):F lattice system if it fails whenever at least one subset of connected failed components occurs which includes failed components connected in the meaning of connected-X. The paper presents some practical examples and the reliability formulas of simple systems using results of consecutive-k-out-of-n:F systems     

A linear bound for sliding-block decoder window size

The author gives an upper bound on the necessary length of a sliding-block decoder window for finite-state codes from arbitrary n -ary data into any constrained system Σ with capacity at least log(n) presented by a graph G with memory m and anticipation a. Specifically, it is shown that the ACH code construction algorithm can be used to construct a code with a sliding-block decoder at rate t:t and with window length m+a+2t, where t is upper-bounded by a linear function of the number of states of G. It is demonstrated that this is the best one can do in the sense that any general upper bound on the decoder window length for finite-state codes into systems Σ with finite memory must grow at least linearly with the number of states of the graph G presenting Σ     

On the number of memories that can be perfectly stored in a neuralnet with Hebb weights

Let {w_ij} be the weights of the connections of a neural network with n nodes, calculated from m data vectors v¹, ···, v^m in {1,-1}ⁿ, according to the Hebb rule. The author proves that if m is not too large relative to n and the v^k are random, then the w_ij constitute, with high probability, a perfect representation of the v^k in the sense that the v ^k are completely determined by the w_ij up to their sign. The conditions under which this is established turn out to be less restrictive than those under which it has been shown that the v^k can actually be recovered by letting the network evolve until equilibrium is attained. In the specific case where the entries of the v^k are independent and equal to 1 or -1 with probability 1/2, the condition on m is that m should not exceed n/0.7 log n     

Distribution of the lifetime of consecutive k-within-m-out-of-n:F systems

The distribution of the lifetime (MTTF) of any consecutive k -within-m-out-of-n:F system, with independent exponentially distributed component lifetimes, is shown to be a convex combination of the distributions (MTTFs) of several convolutions of independent random variables, where each convolution represents a distinct path in the evolution of the system's history, and where in each convolution all but the last random variable is exponential. The last random variable in each convolution is either a zero lifetime or the lifetime of several disjoint consecutive k_i within m_i-out-of-n:F systems in series with each k_i<k, each m_i<m, and each n_i<n. This enables the calculations to proceed recursively. Calculations are facilitated by the symmetric nature of the convex combination     

Minimum distance of logarithmic and fractional partialm-sequences

Two results are presented concerning the partial periods (p-p's) of an m-sequence of period 2ⁿ-1. The first proves the existence of an m-sequence whose p-p's of length approximately (n+d log₂ n) have minimum distance between d and 2d for small d. The second result is of an asymptotic nature and proves that the normalized minimum distance of p-p's whose length is any fraction of the period of the m-sequence, approaches 1/2 as the period of m-sequence tends to infinity     

Pipelined recursive filter with minimum order augmentation

Pipelining is an efficient way for improving the average computation speed of an arithmetic processor. However, for an M-stage pipeline, the result of a given operation is available only M clock periods after initiating the computation. In a recursive filter, the computation of y(n) cannot be initiated before the computations of y(n-1) through y(n-N) are completed. H.B. Voelcker and E.E. Hartquist (1970) and P.M.Kogge and H.S. Stone (1973) independently devised augmentation techniques for resolving the dependence problem in the computation of y(n). However, the augmentation required to ensure stability may be excessively high, resulting in a very complex numerator realization. A technique which results in a minimum order augmentation is presented. The complexity of the resulting filter design is very much lower. Various pipelining architectures are presented. It is demonstrated by an example that when compared to the prototype filter, the augmented filter has a lower coefficient sensitivity and better roundoff noise performance     

Capacity of the Gaussian arbitrarily varying channel

The Gaussian arbitrarily varying channel with input constraint Γ and state constraint Λ admits input sequences x=(x₁,---,X_n) of real numbers with Σx_i²⩽nΓ and state sequences s=(S₁,---,s_n ) of real numbers with Σs_i²⩽nΛ; the output sequence x+s+V, where V=(V₁,---,V_n) is a sequence of independent and identically distributed Gaussian random variables with mean 0 and variance σ². It is proved that the capacity of this arbitrarily varying channel for deterministic codes and the average probability of error criterion equals 1/2 log (1+Γ/(Λ+σ²)) if Λ<Γ and is 0 otherwise     

Index system and separability of constant weight Gray codes

A number system is developed for the conversion of natural numbers to the codewords of the Gray code G(n,k) of length n and weight k, and vice versa. The focus is on the subcode G(n,k) of G(n) consisting of those words of G(n) with precisely k 1-bits, 0<k<n. This code is called the constant weight Gray code of length n and weight k. As an application sharp lower and upper bounds are derived for the value of |i-j|, where i and j are indices of codewords g_i and g_j of G(n,k) such that they differ in precisely 2 m bits     

Sliding block codes between constrained systems

The construction of finite-state codes between constrained systems called sofic systems introduced by R. Karabed and B. Marcus (1988) is continued. It is shown that if Σ is a shift of finite type and S is a sofic system with k/n=h(S )/h(Σ), where h denotes entropy, there is a noncatastrophic finite-state invertible code from Σ to S at rate k:n if Σ and S satisfy a certain algebraic condition involving dimension groups, and Σ and S satisfy a certain condition on their periodic points. Moreover, if S is an almost finite type sofic system, then the decoder can be sliding block     

Energy consumption in multilective and boundary VLSI computations

A VLSI computation is said to be m-way multilective when each input bit is available m times in either space or time or both. The repeated availability of input bits can save computational energy. For a uniswitch m-way multilective computation, where a wire can switch at most once, it is shown that the energy savings can be as much as a factor of m. A multiswitch m-way multilective computation can save up to a factor of √ m switching energy. Tighter energy lower bounds are derived for a circuit with the input/output (I/O) pads located on the border. These boundary computations seem to cost an additional factor ranging from √log n to log n in switching energy. The author extends the energy lower bounds for the multilective case, for a chip with aspect ratio a. The additional energy cost ranges from a factor of √a to a factor of a     

Statistical inference under multiterminal rate restrictions: adifferential geometric approach

A statistical inference problem for a two-terminal information source emitting mutually correlated signals X and Y is treated. Let sequences Xⁿ and Yⁿ of n independent observations be encoded independently of each other into message sets M_X and M_Y at rates R₁ and R ₂ per letter, respectively. This compression causes a loss of the statistical information available for testing hypotheses concerning X and Y. The loss of statistical information is evaluated as a function of the amounts R₁ and R ₂ of the Shannon information. A complete solution is given in the case of asymptotically complete data compression, R₁, R₂→0 as n→∞. It is shown that the differential geometry of the manifold of all probability distributions plays a fundamental role in this type of multiterminal problem connecting Shannon information and statistical information. A brief introduction to the dually coupled e-affine and m-affine connections together with e -flatness and m-flatness is given     

Fast decoding of codes from algebraic plane curves

Improvement to an earlier decoding algorithm for codes from algebraic geometry is presented. For codes from an arbitrary regular plane curve the authors correct up to d*/2-m² /8+m/4-9/8 errors, where d* is the designed distance of the code and m is the degree of the curve. The complexity of finding the error locator is O(n^7/3 ), where n is the length of the code. For codes from Hermitian curves the complexity of finding the error values, given the error locator, is O(n²), and the same complexity can be obtained in the general case if only d*/2-m²/2 errors are corrected     

A coding scheme for m-out-of-n codes

A scheme for the construction of m-out-of-n codes based on the arithmetic coding technique is described. For appropriate values of n, k, and m, the scheme can be used to construct an (n,k) block code in which all the codewords are of weight m. Such codes are useful, for example, in providing perfect error detection capability in asymmetric channels such as optical communication links and laser disks. The encoding and decoding algorithms of the scheme perform simple arithmetic operations recursively, thereby facilitating the construction of codes with relatively long block sizes. The scheme also allows the construction of optimal or nearly optimal m-out-of-n codes for a wide range of block sizes limited only by the arithmetic precision used     

Binary [18,11]2 codes do not exist-Nor do [64,53]2 codes

A binary, linear block code C with block length n and dimension n is commonly denoted by [n, k] or, if its minimum distance is d, by [n, k,d]. The code's covering radius r(C) can be defined as the smallest number r such that any binary column vector of length (n-k) can be written as a sum of r or fewer columns of a parity-check matrix of C. An [n,k] code with covering radius r is denoted by [n,k]r. R.A. Brualdi et al., (1989) showed that l(m,r) is defined to be the smallest n such that an [n,n-m]r code exists. l(m,2) is known for m⩽6, while it is shown by Brualdi et al. that 17⩽l(7,2)⩽19. This lower bound is improved by A.R. Calderbank et al. (1988), where it is shown that [17,10]2 codes do not exist. The nonexistence of [18,11]2 codes is proved, so that l(7,2)=19. l[7.2)=19 is established by showing that [18,11]2 codes do not exist. It is also shown that [64,53]2 codes do not exist, implying that l(11,2)⩾65     

Reliability polynomial for a ring network

A mathematical model is developed for the reliability of a system made up of m unreliable nodes arranged in a ring. The model can be used to calculate the reliability of single-ring networks in which the network recovery mechanism depends on bypassing failed stations, but link signal power margins are inadequate to overcome losses due to more than n bypass switches in series. Computational complexity is 0(n²m+nm²/2) in time, and 0(m²/2) in memory requirements     

Asymptotic performance of M-ary orthogonal modulation ingeneralized fading channels

The asymptotic (M→∞) probability of symbol error P_e,_m for M-ary orthogonal modulation in a Nakagami-m fading channel is given by the incomplete gamma function P(m, mx) where x=In 2/(E_b/N₀) and E_b is the average energy per bit. For large signal-to-noise ratio this leads to a channel where the probability of symbol error varies as the inverse mth power of E_b/N₀. These channels exist for all m⩾1/2. The special case of m=1 corresponds to Rayleigh fading, an inverse linear channel     

A new organic electrooptic crystal:2,6-dibromo-N-methyl-4-nitroaniline (DBNMNA)

A new organic electrooptic crystal, 2,6-dibromo-N-methyl-4-nitroaniline, is reported. The crystal structure was determined by X-ray diffraction (orthorhombic, space group Fdd2, point group mm2, Z=16, a=11.745 Å, b=29.640 Å, c=10.807 Å). The refractive indexes were measured at several wavelengths between 500 and 1100 nm (at 632.8 nm, n_a=1.90, n_b=1.62, and n_c=1.48) and fit the data to a single oscillator Sellmeier equation. A report is presented on measurements of the linear electrooptic effect, yielding the combined coefficients n_a ³r₁₃-n_c³ r₃₃, as well as the first known measurements of the quadratic electrooptic effect in an organic crystal, yielding the coefficients r₄₂ and r₅₁ at 514.5, 632.8, and 810 nm. Good agreement is found between the observed dispersion in the electrooptic coefficients and that predicted by the simple two-level dispersion model     

A construction of non-Reed-Solomon type MDS codes

A construction is presented of long maximum-distance-separable (MDS) codes that are not generalized Reed-Solomon (GRS) type. The construction uses subsets S,|S|=m of a finite field F=GF(q) with the property that no t distinct elements of S add up to some fixed element of F . Large subsets of this kind are used to construct [n=m+2, k=t+1] non-GRS MDS codes over F     

AlInGaAs-AlGaAs strained single-quantum-well diode lasers

Diode lasers with a strained AlInGaAs active layer and AlGaAs confining and cladding layers are reported. Broad-stripe devices were fabricated in graded-index separate-confinement heterostructures grown by organometallic vapor-phase epitaxy on GaAs substrates and containing a single Al_yIn_xGa_1-x-yAs quantum well with 0.12⩽x⩽0.14 and five values of y between 0.05 and 0.17. With increasing Al content the emission wavelength decreases from 890 to 785 nm. The threshold current density J _th is less than 200 A-cm^-2, with one exception, and the differential quantum efficiency ranges from 78 to 81%. A preliminary reliability test was made on a saw-cut, uncoated broad-area device, with an Al_0.17In_0.12Ga_0.71As active layer, that was fabricated without using special precautions to minimize damage and was mounted junction side up. After 20 h of CW operations at a constant current of 1.125 times threshold, J _th has increased by only 3.5%