Higher dimensional orthogonal designs and applications

The concept of orthogonal design is extended to higher dimensions. A properg-dimensional design[d_{ijk cdots upsilon}]is defined as one in which all parallel(g-1)-dimensional layers, in any orientation parallel to a hyper plane, are uncorrelated. This is equivalent to the requirement thatd_{ijk cdots upsilon} in {0, pm x_{1}, cdots , pm x_{t} }, wherex_{1}, cdots , x_{t}are commuting variables, and thatsum_{p} sum_{q} sum_{r} cdots sum_{y} d_{pqr cdots ya} d_{pqr cdots yb} = left( sum_{t} s_{i}x_{i}^{2} right)^{g-1} delta ab,where(s{1}, cdots , s{t})are integers giving the occurrences ofpm x_{1}, cdots , pm x_{t}in each row and column (this is called the type(s_{1}, cdot ,s_{t})^{g-1})and(pqr cdots yz)represents all permutations of(ijk cdots upsilon). This extends an idea of Paul J. Shlichta, whose higher dimensional Hadamard matrices are special cases withx_{1}, cdots , x_{t} in {1,- 1}, (s_{1}, cdots, s_{t})=(g), and(sum_{t}s_{i}x_{i}^{2})=g. Another special case is higher dimensional weighing matrices of type(k)^{g}, which havex_{1}, cdots , x_{t} in {0,1,- 1}, (s_{1}, cdots, s_{t})=(k), and(sum_{t}s_{i}x_{i}^{2})=k. Shlichta found properg-dimensional Hadamard matrices of size(2^{t})^{g}. Proper orthogonal designs of type     

An approach to Ungerboeck coding for rectangular signal sets (Corresp.)

A method of finding good Ungerboeck codes for large rectangular [quadrature amplitude modulation (QAM)] signal sets is described. Using the concept of Euclidean weights due to Ungerboeck, we prove that a2^{n}point basic constellation may be employed to determine exactly the free distance for an Ungerboeck-coded rectangular2^{m}point set, whenm-n-1bits are uncoded and the remaining bits pass through a rate(n-1)/nconvolutionai encoder. It is shown that rate2/3encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered.     

Further classifications of codes meeting the Griesmer bound (Corresp.)

For any(n, k, d)binary linear code, the Griesmer bound says thatn geq sum_{i=0}^{k-1} lceil d/2^{i} rceil, wherelceil x rceildenotes the smallest integergeq x. We consider codes meeting the Griesmer bound with equality. These codes have parametersleft( s(2^{k} - 1) - sum_{i=1}^{p} (2^{u_{i}} - 1), k, s2^{k-1} - sum_{i=1}^{p} 2^{u_{i} -1} right), wherek > u_{1} > cdots > u_{p} geq 1. We characterize all such codes whenp = 2oru_{i-1}-u_{i} geq 2for2 leq i leq p.     

Studies of the inversion and lower level relaxation in pulsed TE CO2discharges

It has generally been supposed that the rise and fall times of the gain in pulsed CO2laser discharges reflect the effective lifetimes of the lower and upper laser levels. Consequently, the ratio of the upper to lower level populationsN_{u}/N_{l}should change during the evolution of the gain, and should be a relatively large number at the gain peak. The problems associated with the determination of this ratio using conventional direct gain measurements are discussed, and a new in-cavity technique is described that has sufficient precision to determine the ratioN_{u}/N_{l}as a function of time. The results obtained unambiguously demonstrate that the lower laser level empties rapidly during the rise time of the gain. This conclusion is further reinforced by some measurements concerning the time variation of the gain in the hot band (01^{1}1 rightarrow 11^{1}0).     

Weight enumerator for second-order Reed-Muller codes

In this paper, we establish the following result. Theorem:A_i, the number of codewords of weightiin the second-order binary Reed-Muller code of length2^mis given byA_i = 0unlessi = 2^{m-1}or2^{m-1} pm 2^{m-l-j}, for somej, 0 leq j leq [m/2], A_0 = A_{2^m} = 1, and begin{equation} begin{split} A_{2^{m-1} pm 2^{m-1-j}} = 2^{j(j+1)} &{frac{(2^m - 1) (2^{m-1} - 1 )}{4-1} } \ .&{frac{(2^{m-2} - 1)(2^{m-3} -1)}{4^2 - 1} } cdots \ .&{frac{(2^{m-2j+2} -1)(2^{m-2j+1} -1)}{4^j -1} } , \ & 1 leq j leq [m/2] \ end{split} end{equation} begin{equation} A_{2^{m-1}} = 2 { 2^{m(m+1)/2} - sum_{j=0}^{[m/2]} A_{2^{m-1} - 2^{m-1-j}} }. end{equation}     

4C6--Concentration and temperature dependences of spin-lattice relaxation times in ruby at helium temperatures: Relaxation in a zero magnetic field

The relaxation times T1for the grouud state levels in ruby were measured in the temperature range 4.2 to 1.6°K for the various concentrations of theCr^{3+}ionsffrom 0.05 to 0.7 percent. The dependenceT_{1}(f)of the formT_{1}^{-1}(f) = T_{1}^{-1}(0) + T_{1}^{-1}(I)f ^{n}withn simeq 2has been obtained for the different transitions. The measurements of relaxation times forpm frac{1}{2} leftrightarrow pm frac{3}{2}transition at zero magnetic field were especially aimed at establishing a form of dependenceT_{1}(f)because of the absence of the cross relaxation effects in this case. The normal temperature dependenceT_{1} propto T_{1}^{-1}has been obtained at all concentrations in comparison with anomalous dependences observed at high concentrations by some researchers.     

Small-signal gain and saturation parameter of a transverse-flow CW oxygen-iodine laser

The small-signal gain and saturation parameter of a transverse-flow CW oxygen-iodine laser have been experimentally obtained for the first time from output power measurements made as a function of the cavity losses without using a CW probe laser. These measurements typically yieldalpha_{0} simeq 0.045m^-1andI_{s} simeq 0.44kW/cm²for a Cl2flow rate of1.4 times 10^{-3}mol/s with an I2flow rate of4 times 10^{-6}mol/s. The dependences of the small-signal gain and saturation parameter have been also found on the Cl2flow rate. These behaviors are qualitatively explained by a simple two-level model.     

Second-harmonic generation in CDA and CD*A

Efficient second-harmonic generation at 5321 Å has been achieved in CDA and CD*A. At 50-MW pump power, loss-free peak power-conversion efficiency of 57 percent and 45 percent were obtained in CDA and CD*A, respectively, The nonlinear optical constant, corrected for multimode fluctuations, was found to bed_{36}(CDA) = d_{36}(CD*A) = (0.96 pm 0.11) times 10^{-9}ESU, which corresponds tod_{36}(CDA) = d_{36}(CD*A) = 0.92 times d_{36}(KDP). In addition, the temperature variation of the birefringenced(n_{2}^{e} - n_{1}^{o})/dThas been measured to be(8.0 pm 0.2) times 10^{-6}degC^-1for CDA and(7.8 pm 0.2) times 10^{-6}degC^-1for CD*A.     

High-radix transforms for Reed-Solomon codes over Fermat primes (Corresp.)

It is shown that a high-radix fast Fourier transform (FFT) with generatorgamma = 3over GF(F_{n}), whereF_{n} = 2^{2}^{n'} + 1is a Fermat prime, can be used for encoding and decoding of Reed-Solomon (RS) codes of length2^{2}^{n}. Such an RS decoder is considerably faster than a decoder using the usual radix 2 FFT. This technique applies most ideally to a 16-error-correcting, 256-symbol RS code of 8 bits being considered currently for space communication applications. This special code can be encoded and decoded rapidly using a high-radix FFT algorithm over GF(F_{3}).     

Hybrid low autocorrelation sequences (Corresp.)

Skew-symmetric sequences of(2n + 1)terms,a_0,a_1,cdots,a_{2n}, are described for which the "merit factor" begin{equation} F_h = frac{biggl[sum_{i=0}^{2n} mid a_i mid biggr] ^2}{ 2 sum_{k=1}^{2n} biggl[ sum_{i=0}^{2n-k} text{sign} (a_i) cdot a_{i+k} biggl] ^2} end{equation} is unusually high.     

Linear feedback rate bounds for regressive channels (Corresp.)

This article presents new tighter upper bounds on the rate of Gaussian autoregressive channels with linear feedback. The separation between the upper and lower bounds is small. We havefrac{1}{2} ln left( 1 + rho left( 1+ sum_{k=1}^{m} alpha_{k} x^{- k} right)^{2} right) leq C_{L} leq frac{1}{2} ln left( 1+ rho left( 1+ sum_{k = 1}^{m} alpha_{k} / sqrt{1 + rho} right)^{2} right), mbox{all rho}, whererho = P/N_{0}W, alpha_{l}, cdots, alpha_{m}are regression coefficients,Pis power,Wis bandwidth,N_{0}is the one-sided innovation spectrum, andxis a root of the polynomial(X^{2} - 1)x^{2m} - rho left( x^{m} + sum^{m}_{k=1} alpha_{k} x^{m - k} right)^{2} = 0.It is conjectured that the lower bound is the feedback capacity.     

Ellipse rotation studies in laser host materials

Using a TEM00qnear Gaussian mode ruby laser system we report the first experimental measurements of intensity induced changes of optical polarization (ellipse rotation) in a cubic crystalline medium, YAG, for which we obtain the nonlinear susceptibilitieschi_{3}^{1221} (- omega, omega, omega, -omega) = 6.34 times 10^{-15}ESU andfrac{1}{2} (chi_{3}^{1111} + chi_{3}^{1221} - 2chi_{3}^{1212}) = 7.18 times 10^{-15}ESU, accurate to better than ±7 percent relative tochi_{3}^{1221} (- omega,omega,omega, -omega)for liquid CS2. These values are compared with further results obtained for fused quartz and two laser glasses. Moreover, by time resolving the ellipse rotation data we demonstrate the capability to plot ellipse rotation versus input power on a single laser shot, thus increasing the practical feasibility of the technique and introducing the possibility of resolving transient contributions to the measurement.     

Small signal gain in the CO-air-He laser

The small-signal single-pass gain for 24 individual CO laser lines has been measured for a flowing CO-air-He laser. The vibrational-rotationalP(J)_{upsilon,upsilon'}lines considered were for rotational transitions fromJ = 10to 16 and upper vibrational levels fromupsilon = 7to 11. The maximum gain was observed for theP(15)_{10,9}laser line, with an average logarithmic energy gain of2.633 times 10^{-3}cm^-1.     

Multiple-burst error-correcting cyclic product codes (Corresp.)

LetCbe the cyclic product code ofpsingle parity check codes of relatively prime lengthsn_{1}, n_{2},cdots , n_{p} (n_{1} < n_{2} < cdots < n_{p}). It is proven thatCcan correct2^{P-2}+2^{p-3}-1bursts of lengthn_{1}, andlfloor(max{p+1, min{2^{p-s}+s-1,2^{p-s}+2^{p-s-1}}}-1)/2rfloorbursts of lengthn_{1}n_{2} cdots n_{s} (2leq s leq p-2). Forp=3this means thatCis double-burst-n_{1}-correcting. An efficient decoding algorithm is presented for this code.     

Performance of anM-ary orthogonal communication system using stationary stochastic signals

This paper considers the performance of a communication system which transmits forTseconds the real part of a sample function of one ofMstationary complex Gaussian processes whose spectral densities are all frequency translations of the functionS_{xi (f). At the receiver white Gaussian noise of one-sided densityN_{0}is added. The center frequencies of the processes are assumed to be sufficiently separated that theMcovariance functions are orthogonal overT. Exponently tight bounds are obtained for the error probability of the maximum likelihood receiver. It is shown that the error probability approaches zero exponentially withTfor all ratesR = (ln M)/Tup toC= int_{-infty}^{infty} [S_{xi (f)/N_{0}] df - int_{- infty}^{infty} ln [1 + S_{xi}(f)/N_{0}] dfwhich is shown to be the channel capacity. Similar results are obtained for the case of stochastic signals with specular components.     

The inhibition of negative resistance dipole waves and domains in n-GaAs

Expressions describing the growth and propagation of infinitesimal space-charge waves in the presence of a differential negative resistance are derived, and conditions for the inhibition of dipole waves and domains inn-GaAs of resistivity greater than 10Ω-cm at 300°K are obtained. Thermoelectric effects are estimated and found to be small for carrier densities much less than 10¹⁵cm^-3. Taking the negative differential resistivity to be larger than the ohmic resistivity by a factor of 10²leads to 1) condition for inhibition of dipole waves:nl^{2} lsim 10^{9}cm^-12) condition for inhibition of stable domains:nl lsim 10^{12}cm^-2whenn= electron density andl= specimen length. The relevance of these results to explaining the phenomena observed in amplifying crystals is pointed out. It is suggested that the condition for amplification is(frac{10^{9}}{n})^{1/2} lsim l lsim frac{10^{12}}{n}.     

The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

It is shown thatsqrt[8]{2}is an element of order2^{n+4}inGF(F_{n}), whereF_{n}=2^{2^{n}}+1is a Fermat prime forn=3,4. Hence it can be used to define a fast Fourier transform (FFT) of as many as2^{n+4}symbols inGF(F_{n}). Sincesqrt[8]{2}is a root of unity of order2^{n+4}inGF(F_{n}), this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the casen=2, wheresqrt{2}is inGF(F_{2})=GF(17), is given.     

Self-Orthogonal Convolutional Coding for the DPCM-AQB Speech Encoder

A fixed-tap differential pulse code modulation (DPCM) system with a robust backward-adaptive Jayant quantizer is investigated for speech encoding at 16-40 kbits/s using binary phase shift keying over an additive white Gaussian noise channel. The performance of this system becomes unacceptable as the channel bit error rate(P_{b})approaches 10^-2. Using high-rate, long constraint length, self-orthogonal convolutional codes, the DPCM system performance is much-improved for10^{-4} < P_{b} < 10^{-2}depending on the transmitted data rate. The use of high-rate(n - 1)/n, n = 2,3,4,, and 5 codes minimizes the number of bits allocated to channel coding, and decoding complexity is reduced by employing self-orthogonal codes which admit threshold decoding. Subjectively, while there is additional quantization noise with channel coding, the irritating popping and squeaking sounds due to channel errors are eliminated.     

Dual product constructions of Reed - Muller type codes (Corresp.)

Various linear and nonlinearR(r,m)codes having parameters(2^{m}, 2^{k}, 2^{m-r})withk=sum_{i=0}^{r}left(^{m}_{i}right)are constructed fromR(r,q)andR(r,p)codes,m=p+q. A dual construction forR(m-r,m)codes fromR(p-r,p)andR(q-r,q)codes is also presented,m=p+q. As a simple corollary we have that the number of nonequivalentR(r,m)codes is at least exponential in the length (forr>1). ForR(m-r,m)codes, the lower bound is doubly exponential in the length (forr>1).     

Measurements of anomalous gain coefficients in transversely excited CO2lasers

We have made a detailed study of the gain coefficients of more than 150 transitions in the [00^{0}1-10^{0}0]_{I,II}and [00^{0}2-10^{0}1]_{I,II}bands of CO2. Measurements were made in a transversely excited discharge at pressures of 80 and 425 torr. Many lines were found to have anomalously high gain coefficients, but all the anomalies are accurately predicted by a model which takes into account the effect of overlapping transitions. Our results indicate that the majority of transitions in an atmospheric pressure discharge have anamalous gain coefficients under typical discharge conditions. This fact has not been generally recognized, and the neglect of overlapping transitions can lead to errors in determining rotational temperatures.