Galois field arithmetic over GF(p/sup m/) for high-speed/low-power error-control applications

This paper presents a very large-scale integration implementation of Galois field arithmetic for high-speed error-control coding applications that is based on the field GF(p/sup m/) with m a small integer such as 2 or 3 and p a prime of sufficient value to generate the required field size. In this case, the Galois field arithmetic operations of addition, multiplication, and inversion are based on architectures using blocks that perform integer arithmetic modulo p. These integer arithmetic operations modulo p have previously been implemented with low delay power products through the use of one hot coding and barrel shifters circuits based on transistor arrays. In this paper, the same one hot coding and barrel shifters circuits are used to construct circuits that implement addition, multiplication, and inversion over GF(p/sup m/). The circuits for GF(p/sup m/) addition and multiplication with p/spl ne/2, achieve a lower power-delay product than designs based on GF(2/sup m/). Also, the architecture for GF(p/sup m/) inversion can be efficiently implemented when m=2 or m=3.     

Entropy and the law of small numbers

Two new information-theoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary information-theoretic techniques it is shown that, when S/sub n/=/spl Sigma//sub i=1//sup n/X/sub i/ is the sum of the (possibly dependent) binary random variables X/sub 1/,X/sub 2/,...,X/sub n/, with E(X/sub i/)=p/sub i/ and E(S/sub n/)=/spl lambda/, then D(P(S/sub n/)/spl par/Po(/spl lambda/)) /spl les//spl Sigma//sub i=1//sup n/p/sub i//sup 2/+[/spl Sigma//sub i=1//sup n/H(X/sub i/)-H(X/sub 1/,X/sub 2/,...,X/sub n/)] where D(P(S/sub n/)/spl par/Po(/spl lambda/)) is the relative entropy between the distribution of S/sub n/ and the Poisson (/spl lambda/) distribution. The first term in this bound measures the individual smallness of the X/sub i/ and the second term measures their dependence. A general method is outlined for obtaining corresponding bounds when approximating the distribution of a sum of general discrete random variables by an infinitely divisible distribution. Second, in the particular case when the X/sub i/ are independent, the following sharper bound is established: D(P(S/sub n/)/spl par/Po(/spl lambda/))/spl les/1//spl lambda/ /spl Sigma//sub i=1//sup n/ ((p/sub i//sup 3/)/(1-p/sub i/)) and it is also generalized to the case when the X/sub i/ are general integer-valued random variables. Its proof is based on the derivation of a subadditivity property for a new discrete version of the Fisher information, and uses a recent logarithmic Sobolev inequality for the Poisson distribution.     

A fast algorithm for determining the minimal polynomial where of a sequence with period 2p/sup n/ over GF (q)

A fast algorithm is presented for determining the linear complexity and the minimal polynomial of a sequence with period 2p/sup n/ over GF (q), where p and q are odd prime, and q is a primitive root (mod p/sup 2/). The algorithm uses the fact that in this case the factorization of x/sup 2p(n)/-1 is especially simple.     

Compression mappings on primitive sequences over Z/(p/sup e/)

Let Z/(p/sup e/) be the integer residue ring with odd prime p/spl ges/5 and integer e/spl ges/2. For a sequence a_ over Z/(p/sup e/), there is a unique p-adic expansion a_=a_/sub 0/+a_/spl middot/p+...+a_/sub e-1//spl middot/p/sup e-1/, where each a_/sub i/ is a sequence over {0,1,...,p-1}, and can be regarded as a sequence over the finite field GF(p) naturally. Let f(x) be a primitive polynomial over Z/(p/sup e/), and G'(f(x),p/sup e/) the set of all primitive sequences generated by f(x) over Z/(p/sup e/). Set /spl phi//sub e-1/ (x/sub 0/,...,x/sub e-1/) = x/sub e-1//sup k/ + /spl eta//sub e-2,1/(x/sub 0/, x/sub 1/,...,x/sub e-2/) /spl psi//sub e-1/(x/sub 0/,...,x/sub e-1/) = x/sub e-1//sup k/ + /spl eta//sub e-2,2/(x/sub 0/,x/sub 1/,...,x/sub e-2/) where /spl eta//sub e-2,1/ and /spl eta//sub e-2,2/ are arbitrary functions of e-1 variables over GF(p) and 2/spl les/k/spl les/p-1. Then the compression mapping /spl phi//sub e-1/:{G'(f(x),p/sup e/) /spl rarr/ GF(p)/sup /spl infin// a_ /spl rarr/ /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) is injective, that is, a_ = b_ if and only if /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) = /spl phi//sub e-1/(b_/sub 0/,...,b_/sub e-1/) for a_,b_ /spl isin/ G'(f(x),p/sup e/). Furthermore, if f(x) is a strongly primitive polynomial over Z/(p/sup e/), then /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) = /spl psi//sub e-1/(b_/sub 0/,...,b_/sub e-1/) if and only if a_ = b_ and /spl phi//sub e-1/(x/sub 0/,...,x/sub e-1/) = /spl psi//sub e-1/(x/sub 0/,...,x/sub e-1/) for a_,b_ /spl isin/ G'(f(x),p/sup e/).     

Multistage list decoding of generalized Reed-Solomon codes over Galois rings

A two-stage list decoder for generalized Reed-Solomon codes over GR (p/sup l/,m) that can exceed the Guruswami-Sudan decoding radius t/sub GS/ with significant probability for p=2, l/spl ges/2 and m=1 was recently proposed. It makes a distinction between error values which are units and those which are zero divisors in order to exceed t/sub GS/. This letter presents an extension of that approach by exploiting the fact that each element of GR (p/sup l/,m) has a unique p-adic expansion, culminating in a multistage decoder that outperforms the two-stage decoder.     

A class of 1-generator quasi-cyclic codes

If R = F/sub q/[x/spl rceil/]/(x/sup m/ - 1), S = F/sub qn/[x]/(x/sup m/ - 1), we define the mapping a_(x) /spl rarr/ A(x) =/spl sigma//sub 0//sup n-1/a/sub i/(x)/spl alpha//sub i/ from R/sup n/ onto S, where (/spl alpha//sub 0/, /spl alpha//sub i/,..., /spl alpha//sub n-1/) is a basis for F/sub qn/ over F/sub q/. This carries the q-ray 1-generator quasicyclic (QC) code R a_(x) onto the code RA(x) in S whose parity-check polynomial (p.c.p.) is defined as the monic polynomial h(x) over F/sub q/ of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of x/sub m/ 1 over F/sub q/ and F/sub qn/ are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal in S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for F/sub qn/ over F/sub q/, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes.     

Propagation in Twisted Square Waveguide

The problem of propagation of TE modes in twisted rectangular waveguides has been solved except for the case where one of the propagating modes becomes degenerate. The purpose of this paper is to show how to obtain a solution for degenerate modes in a twisted rectangular waveguide, with emphasis on the particular case of the square waveguide, for which the lowest order mode is degerate. It is shown that the propagation constant for the twisted square waveguide can be expressed as an asymptotic series, the first term being the propagation constant for a straight square waveguide and the first order correction term being of order 1/ L where L is the distance in which the guide makes one full rotation. The propagation constant for a nondegenerate mode in a twisted rectangular waveguide, on the other hand, can be expressed in a similar manner except that the first-order correction term is of order 1/ L/sup2/. Some comments are offered on the nature of the transition when the propagating mode is almost degenerate.     

Frequency hopping sequences with optimal partial autocorrelation properties

We classify some p/sup k/-ary (p prime, k integer) generalized m-sequences and generalized Gordon-Mills-Welch (GMW) sequences of period p/sup 2k/-1 over a residue class ring R=GF(p)[/spl xi/]/(/spl xi//sup k/) having optimal partial Hamming autocorrelation properties. In frequency hopping (FH) spread-spectrum systems, these sequences are useful for synchronizing process. Suppose, for example, that a transmitting p/sup k/-ary FH patterns of period p/sup 2k/-1 are correlated at a receiver. Usually, the length of a correlation window, denoted by L, is shorter than the pattern's overall period. In that case, the maximum value of the out-of-phase Hamming autocorrelation is lower-bounded by /spl lceil/L/p/sup k/+1/spl rceil/ but the classified sequences achieve this bound with equality for any positive integer L.     

Capacitance Definitions for Parametric Operation (Correspondence)

The purpose of this note is to reconcile the various definitions of nonlinear or equivalent time-varying capacitance which appear in the literature concerned with parametric devices. The problem considered is one of a nonlinear reactive element, let us say a capacitance, which is pumped by strong pump source at frequency f/sub p/ and which couples two circuit modes at frequencies f/sub s and f/sub i/, usually called the signal and idling frequencies. For parametric operation we demand either f/sub p/=f/sub l/ + f/sub s/ or f/sub p/ = f/sub l - f/sub s/.     

Comments on "Ridged Waveguides with Inhomogeneous Dielectric Slab Loading" (Letters)

In the above paper/sup1/, Magerl derives dispersion equations for TE/sub m0/ modes of ridged waveguide with inhomogeneous dielectric-slab loading. The existence of TE modes was evidently assumed a priori (a seemingly natural assumption); unfortunately, true TE modes do not exist in this waveguide (with a singuIar exception to be noted).     

Asymptotic properties on codeword lengths of an optimal FV code for general sources

This correspondence is concerned with asymptotic properties on the codeword length of a fixed-to-variable length code (FV code) for a general source {X/sup n/}/sub n=1//sup /spl infin// with a finite or countably infinite alphabet. Suppose that for each n /spl ges/ 1 X/sup n/ is encoded to a binary codeword /spl phi//sub n/(X/sup n/) of length l(/spl phi//sub n/(X/sup n/)). Letting /spl epsiv//sub n/ denote the decoding error probability, we consider the following two criteria on FV codes: i) /spl epsiv//sub n/ = 0 for all n /spl ges/ 1 and ii) lim sup/sub n/spl rarr//spl infin///spl epsiv//sub n/ /spl les/ /spl epsiv/ for an arbitrarily given /spl epsiv/ /spl isin/ [0,1). Under criterion i), we show that, if X/sup n/ is encoded by an arbitrary prefix-free FV code asymptotically achieving the entropy, 1/nl(/spl phi//sub n/(X/sup n/)) - 1/nlog/sub 2/ 1/PX/sup n/(X/sup n/) /spl rarr/ 0 in probability as n /spl rarr/ /spl infin/ under a certain condition, where P/sub X//sup n/ denotes the probability distribution of X/sup n/. Under criterion ii), we first determine the minimum rate achieved by FV codes. Next, we show that 1/nl(/spl phi//sub n/(X/sup n/)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.     

Improved-performance, InGaAs/InGaAsP (/spl lambda/=980 nm) asymmetric broad-waveguide diode lasers via waveguide-core doping

Doping the waveguide core (p=2/spl times/10/sup 17/ cm/sup -1/) in asymmetric-waveguide InGaAs/InGaAsP, two-quantum-well diode lasers (/spl lambda/=980 nm) raises the injection efficiency to 90% and decreases the threshold-current density, J/sub th/. For 2 mm long, 100 /spl mu/m wide stripe, uncoated chips J/sub th/ decreases from /spl sim/188 A/cm/sup 2/ to /spl sim/150 A/cm/sup 2/. High characteristic temperatures for J/sub th/ and the slope efficiency are obtained: T/sub 0/=215K and T/sub 1/=600K.     

Transient Electromagnetic Guided Wave Propagation in Moving Media

An analytical study of the influence of moving media on the propagation of transient electromagnetic modal waves in dispersive waveguides is presented. The response to impulsive excitation is determined in exact closed form and used to demonstrate that the nature of the pulse distortion differs in each of the three cases, 1) 0 < u < c /spl equiv/ (/spl epsi//spl mu/)/sup -1/2/, 2) v = c, and 3) c < v < c/sub 0/ /spl equiv/ (/spl epsi//sub 0//spl mu//sub 0/)/sup -1/2/ (v = speed of the medium). An expression is derived from which the pulse waveform generated by an input of arbitrary form can be readily determined when 0 < v < c if the transient response to a similar input is known for v = 0. An untabulated Laplace transform pair is derived and used to determine the unusual pulse distortion in case 3) which shows a markedIy discontinuous change from the pulse distortion in case 2). The theory illuminates a singular circumstance in which the Lorentz transformation is consistent with the "speed of light" differing from one inertial reference frame to another.     

Lossless and near-lossless source coding for multiple access networks

A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences {X/sub i/}/sub i=1//sup /spl infin// and {Y/sub i/}/sub i=1//sup /spl infin// is drawn independent and identically distributed (i.i.d.) according to joint probability mass function (p.m.f.) p(x,y); the encoder for each source operates without knowledge of the other source; the decoder jointly decodes the encoded bit streams from both sources. The work of Slepian and Wolf describes all rates achievable by MASCs of infinite coding dimension (n/spl rarr//spl infin/) and asymptotically negligible error probabilities (P/sub e//sup (n)//spl rarr/0). In this paper, we consider the properties of optimal instantaneous MASCs with finite coding dimension (n相似文献   

Prime-length real-valued polynomial residue division algorithms

One class of efficient algorithms for computing a discrete Fourier transform (DFT) is based on a recursive polynomial factorization of the polynomial 1-z/sup -N/. The Bruun algorithm is a typical example of such algorithms. Previously, the Bruun algorithm, which is applicable only when system lengths are powers of two in its original form, is generalized and modified to be applicable to the case when the length is other than a power of two. This generalized algorithm consists of transforms T/sub d,f/ with prime d and real f in the range 0/spl les/f<0.5. T/sub d,0/ computes residues X(z)mod(1-z/sup -2/) and X(z)mod(1-2 cos(/spl pi/k/d)z/sup -1/+z/sup -2/), k=1, 2, ..., d-1, and T/sub d,f/ (f /spl ne/0) computes residues X(z)mod(1-2cos(2/spl pi/(f+k)/d)z/sup -1/+z/sup -2/), k=0, 1, ..., d-1 for a given real signal X(z) of length 2d. The purpose of this paper is to find efficient algorithms for T/sub d,f/. First, polynomial factorization algorithms are derived for T/sub d,0/ and T/sub d,1/4/. When f is neither 0 nor 1/4, it is not feasible to derive a polynomial factorization algorithm. Two different implementations of T/sub d,f/ for such f are derived. One implementation realizes T/sub d,f/ via a d-point DFT, for which a variety of fast algorithms exist. The other implementation realizes T/sub d,f/ via T/sub d, 1/4/, for which the polynomial factorization algorithm exists. Comparisons show that for d/spl ges/5, these implementations achieve better performance than computing each output of T/sub d,f/ separately.     

The influence of oxygen partial pressure on growth of the (Hg,Re)-1223 intergrain junction

Hg/sub 0.82/Re/sub 0.18/Ba/sub 2/Ca/sub 2/Cu/sub 3/O/sub 8+/spl delta// polycrystalline samples were successfully obtained by using different oxygen partial pressure in the annealing treatment of the precursor ceramic. The doping state was confirmed by X-ray powder diffraction pattern analysis and by observing distinct thermopower values at room temperature. Also, the intergrain regions have shown an improvement in the critical current density when using the precursor preparation with 10% O/sub 2/ and 90% Ar (optimal doped). The optimal doped sample has presented the highest /spl alpha/ exponent of the J/sub c//spl prop/[1-(T/T/sub c/)/sup 2/]/sup /spl alpha// dependence. For the case of (Hg,Re)-1223 polycrystalline superconductor applications, the /spl alpha/ exponent can be used as a junction quality parameter.     

Investigation of a periodically segmented waveguide Fabry-Pe/spl acute/rot interferometer for use as a chemical/biosensor

We present a periodically segmented waveguide Fabry-Pe/spl acute/rot interferometer (PSW-FPI) intended to be used for tagless real-time chemical/ biological sensing through bulk-material interaction. The differential sensor detects changes in the refractive index (RI) of a sample regardless of its absolute RI value. Experiments with a series of sucrose solutions of various concentrations are compared with theoretical results, and a very good match is found between the two. The theoretical sensitivity limit for a 50-dB-signal-to-noise-ratio (SNR) measurement system is estimated as /spl delta/n=3/spl middot/10/sup -7/. For a 29-dB-SNR measurement system, a measured sensitivity limit of /spl delta/n=4/spl middot/10/sup -5/ is comparable with the previously reported sensitivities of label-free real-time optical biosensors (2/spl middot/10/sup -5/-5/spl middot/10/sup -5/). However, the total required sensing length (720 /spl mu/m) of our PSW-FPI sensor is much shorter than that of the previously reported devices (9-20 mm).     

Theory of Direct-Coupled-Cavity Filters

A new theory is presented for the design of direct-coupled-cavity filters in transmission line or waveguide. It is shown that for a specified range of parameters the insertion-loss characteristic of these filters in the case of Chebyshev equal-ripple characteristic is given very accurately by the formula P/sub 0/ / /P/sub L/ = 1+h/sup 2/T/sub n//sup 2/[/spl omega//sub 0/ / /spl omega/ sin(/spl pi/ /spl omega/ / /spl omega//sub 0/) / sin/spl theta//sub 0/'] where h defines the ripple level, T/sub n/ is the first-kind Chebyshev polynomial of degree n, /spl omega/ / /spl omega//sub 0/ is normalized frequency, and /spl theta//sub 0/' is an angle proportional to the bandwidth of a distributed lowpass prototype filter. The element values of the direct-coupled filter are related directly to the step impedances of the prototype whose values have been tabulated. The theory gives close agreement with computed data over a range of parameters as specified by a very simple formula. The design technique is convenient for practical applications.     

Cutoff frequencies in diffusion-type power-law profile waveguides

The cutoff frequencies V_c0 of strongly asymmetric planar waveguides with diffusion-type power-law refractive-index profiles g(X)=X^p are found to be the solutions of J_-ν (2νV_c0)=0 with ν=1/(p+2). To a high degree of accuracy, the cutoff frequencies can be approximated by linear functions of the profile parameter p. The effect of the cover, as well as that of the refractive-index gradient in the case of TM modes, can accurately be taken into account by simple perturbation corrections