An approximation to the weight distribution of binary primitive BCH codes with designed distances 9 and 11 (Corresp.)

Recently Kasami {em et al.} presented a linear programming approach to the weight distribution of binary linear codes [2]. Their approach to compute upper and lower bounds on the weight distribution of binary primitive BCH codes of length2^{m} - 1withm geq 8and designed distance2t + 1with4 leq t leq 5is improved. From these results, the relative deviation of the number of codewords of weightjleq 2^{m-1}from the binomial distribution2^{-mt} left( stackrel{2^{m}-1}{j} right)is shown to be less than 1 percent for the following cases: (1)t = 4, j geq 2t + 1andm geq 16; (2)t = 4, j geq 2t + 3and10 leq m leq 15; (3)t=4, j geq 2t+5and8 leq m leq 9; (4)t=5,j geq 2t+ 1andm geq 20; (5)t=5, j geq 2t+ 3and12 leq m leq 19; (6)t=5, j geq 2t+ 5and10 leq m leq 11; (7)t=5, j geq 2t + 7andm=9; (8)t= 5, j geq 2t+ 9andm = 8.     

Some results on the minimum weight of primitive BCH codes (Corresp.)

For1 leq i leq m - s- 2and0 leq s leq m -2i, the intersection of the binary BCH code of designed distance2 ^{m-s-1} - 2 ^{m-s-t-1} - 1and length2^m - 1with the shortened(s + 2)th-order Reed-Muller code of length2^m -- 1has codewords of weight2^{m-s-1} - 2^{m-s-t-1} - 1.     

An upper bound to the capacity of the band-limited Gaussian autoregressive channel with noiseless feedback

Upper bounds to the capacity of band-limited Gaussianmth-order autoregressive channels with feedback and average energy constraintEare derived. These are the only known hounds on one- and two-way autoregressive channels of order greater than one. They are the tightest known for the first-order case. In this case letalpha_1be the regression coefficient,sigma^2the innovation variance,Nthe number of channel iterations per source symbol, ande = E/N; then the first-order capacityC^1is bounded by begin{equation} C^1 leq begin{cases} frac{1}{2} ln [frac{e}{sigma^2}(1+ mid alpha_1 mid ) ^ 2 +1], & frac{e}{sigma^2} leq frac{1}{1- alpha_1^2} \ frac{1}{2} ln [frac{e}{sigma^2} + frac{2mid alpha_1 mid}{sqrt{1-alpha_1^2}} sqrt{frac{e}{simga^2}} + frac{1}{1-alpha_1^2}], & text{elsewhere}.\ end{cases} end{equation} This is equal to capacity without feedback for very low and very highe/sigma^2and is less than twice this one-way capacity everywhere.     

When does the modular distance induce a metric in the binary case? (Corresp.)

The modular distance induces a metric if and only if the nonadjacent form of the modulusMhas one of the following forms:1) 2^{n}+2^{n-2} pm 2^{i}, wheren-igeq 4; 2) 2^{n} - 2^{j} pm 2^{i}, where2 leq n -j leq 5andj-igeq 2; 3) 2^{n} pm 2^{j}, wheren -j geq 2; 4) 2^{n}.     

A theorem on the entropy of certain binary sequences and applications--I

In this, the first part of a two-part paper, we establish a theorem concerning the entropy of a certain sequence of binary random variables. In the sequel we will apply this result to the solution of three problems in multi-user communication, two of which have been open for some time. Specifically we show the following. LetXandYbe binary randomn-vectors, which are the input and output, respectively, of a binary symmetric channel with "crossover" probabilityp_0. LetH{X}andH{ Y}be the entropies ofXandY, respectively. Then begin{equation} begin{split} frac{1}{n} H{X} geq h(alpha_0), qquad 0 leq alpha_0 &leq 1, Rightarrow \ qquad qquad &qquad frac{1}{n}H{Y} geq h(alpha_0(1 - p_0) + (1 - alpha_0)p_0) end{split} end{equation} whereh(lambda) = -lambda log lambda - (1 - lambda) log(l - lambda), 0 leq lambda leq 1.     

An estimate of the variation of a band-limited process

Upper and lower bounds are established for the mean-square variation of a stationary processX(t)whose power spectrum is bounded byomega_{c}, in terms of its average powerP_{0}and the average powerP_{1}of its derivative. It is shown thatleft( frac{2}{pi} right)^{2} P_{1} tau^{2} leq E {|X(t+tau )-X(t)|^{2}} leq P_{1} tau^{2} leq omega_{c}^{2}P_{0}tau^{2}where the upper bounds are valid for anytauand the lower bound fortau < pi / omega_{c}. These estimates are applied to the mean-square variation of the envelope of a quasi-monochromatic process.     

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.     

On the covering radius of binary codes (Corresp.)

Upper bounds on the covering radius of binary codes are studied. In particular it is shown that the covering radiusr_{m}of the first-order Reed-Muller code of lenglh2^{m}satisfies2^{m-l}-2^{lceil m/2 rceil -1} r_{m} leq 2^{m-1}-2^{m/2-1}.     

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.     

Mutual information of the white Gaussian channel with and without feedback

The following model for the white Gaussian channel with or without feedback is considered: begin{equation} Y(t) = int_o ^{t} phi (s, Y_o ^{s} ,m) ds + W(t) end{equation} wheremdenotes the message,Y(t)denotes the channel output at timet,Y_o ^ {t}denotes the sample pathY(theta), 0 leq theta leq t. W(t)is the Brownian motion representing noise, andphi(s, y_o ^ {s} ,m)is the channel input (modulator output). It is shown that, under some general assumptions, the amount of mutual informationI(Y_o ^{T} ,m)between the messagemand the output pathY_o ^ {T}is directly related to the mean-square causal filtering error of estimatingphi (t, Y_o ^{t} ,m)from the received dataY_o ^{T} , 0 leq t leq T. It follows, as a corollary to the result forI(Y_o ^ {T} ,m), that feedback can not increase the capacity of the nonband-limited additive white Gaussian noise channel.     

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.     

Channel waveguides in glass via silver-sodium field-assisted ion exchange

Multimode channel waveguides were formed by field-assisted diffusion of Ag+ ion from vacuum-evaporated Ag films, into a sodium aluminosilicate glass reported to yield high diffusion rates for alkali ions. Two-dimensional index profiles of channel waveguides formed by diffusion from a strip aperture were controlled by means of diffusion time, temperature, and electric field. The diffusion equation for diffusion through a strip aperture in the presence of a one-dimensional electric field was solved. Its solution was in agreement with measured concentration profiles:frac{C(x,y,t)}{C_{0}} = frac{1}{2} { erf (frac{a - x}{2sqrt{Dt}}) + erf (frac{a + x}{2sqrt{Dt}})}.frac{1}{2} { erfc (frac{y - muE_{y}t}{2sqrt{Dt}}) + e^{(yE_{y}/D)} erfc (frac{y + muE_{y}t}{2sqrt{Dt}})}Diffusion coefficients in this aluminosilicate glass were determined to beD =(2.41 times 10^{-13}) (frac{m^{2}}{s})).exp (frac{-3.1 times 10^{4}frac{J}{mol}}{RT})Diffusion coefficients were higher (between 150°C and 300°C) than those of a low-iron soda-lime silicate glass "standard" also studied, for which diffusion coefficients wereD =(3.28 times 10^{-13} (frac{m^{2}}{s})).exp (frac{-3.6 times 10^{4}}{RT} (frac{J}{mol}))This difference in diffusion coefficients is due to the higher activation energy of diffusion in the soda-lime silicate glass. The Gladstone-Dale relation was used to calculate the maximum possible refractive index change via Ag+-Na+ ion-exchange for each type of glass. The maximum index change in the sodium aluminosilicate glass is found to be about 65 percent of that in the soda-lime silicate glass.     

On the weight distribution of the coset leaders of the first-order Reed - Muller code (Corresp.)

Letalpha_{n}denote the number of cosets with minimum weightnof the(2^{m}, m + 1)Reed-Muller code. Thealpha_{n}for2^{m-2} leq n < 2^{m-2} + 2^{m - 4}is determined.     

Some integrals involving theQ_Mfunction (Corresp.)

Some integrals are presented that can be expressed in terms of theQ_Mfunction, which is defined as begin{equation} Q_M(a,b) = int_b^{infty} dx x(x/a)^{M-1} exp (- frac{x^2 + a^2}{2}) I_{M-1}(ax), end{equation} whereI_{M-1}is the modified Bessel function of orderM-1. Some integrals of theQ_Mfunction are also evaluated.     

On certain projective geometry codes (Corresp.)

LetVbe an(n, k, d)binary projective geometry code withn = (q^{m}-1)/(q - 1), q = 2^{s}, andd geq [(q^{m-r}-1)/(q - 1)] + 1. This code isr-step majority-logic decodable. With reference to the GF(q^{m}) = {0, 1, alpha , alpha^{2} , cdots , alpha^{n(q-1)-1} }, the generator polynomialg(X), ofV, hasalpha^{nu}as a root if and only ifnuhas the formnu = i(q - 1)andmax_{0 leq l < s} W_{q}(2^{l} nu) leq (m - r - 1)(q - 1), whereW_{q}(x)indicates the weight of the radix-qrepresentation of the numberx. LetSbe the set of nonzero numbersnu, such thatalpha^{nu}is a root ofg(X). LetC_{1}, C_{2}, cdots, C_{nu}be the cyclotomic cosets such thatSis the union of these cosets. It is clear that the process of findingg(X)becomes simpler if we can find a representative from eachC_{i}, since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases ofm-r = 2, withs = 2, 3, andm-r=3, withs=2.     

On the inequivalence of generalized Preparata codes

Ifmis odd andsigma /in Aut GF(2^{m})is such thatx rightarrow x^{sigma^{2}-1}is1-1, there is a[2^{m+1}-1,2^{m+l}-2m-2]nonlinear binary codeP(sigma)having minimum distance 5. All the codesP(sigma)have the same distance and weight enumerators as the usual Preparata codes (which rise asP(sigma)whenx^{sigma}=x^{2}). It is shown thatP(sigma)andP(tau)are equivalent if and only iftau=sigma^{pm 1}, andAut P(sigma)is determined.     

Thermionic current in a parallel-plane diode

An approximate formula is developed for the current of a parallel-plane diode including the effects of initial velocities of emission. For an oxide-coated cathode (T = 1000°K) the approximate result is:J = 2.33 times 10^{-6} frac{V^{3/2}}{x^{2}} cdot[1 + frac{11.4(J_{s}x^{2})^{1/4}}{V^{3/4}} + frac{3.22(J_{s}x^{2})^{1/8}}{V^{3/2}}]where,J, Js, andxare in suitable units such that Jx²andJ_{s}x^{2}are in amperes and V in volts. A comparison is made between this result, Child's 3/2 power solution, and the Epstein-Fry-Langmuir solution. The result given above being an explicit solution of the current is particularly advantageous over prior approximate solutions.     

An upper bound for codes for the noisy two-access binary adder channel (Corresp.)

Using earlier methods a combinatorial upper bound is derived for|C|. cdot |D|, where(C,D)is adelta-decodable code pair for the noisy two-access binary adder channel. Asymptotically, this bound reduces toR_{1}=R_{2} leq frac{3}{2} + elog_{2} e - (frac{1}{2} + e) log_{2} (1 + 2e)= frac{1}{2} - e + H(frac{1}{2} - e) - frac{1}{2}H(2e),wheree = lfloor (delta - 1)/2 rfloor /n, n rightarrow inftyandR_{1}resp.R_{2}is the rate of the codeCresp.D.     

Bounds for truncation error in sampling expansions of band-limited signals

Forf(t)a real-valued signal band-limited to- pi r leq omega leq pi r (0 < r < 1)and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error whenf(t)is approximated at a generic timetby an appropriate selection ofN_{1} + N_{2} + 1terms from its Shannon sampling series expansion, the latter expansion being associated with the full band[-pi, pi]and thus involving samples offtaken at the integer points. Results are presented for two cases: 1) the Fourier transformF(omega)is such that|F(omega)|^{2}is integrable on[-pi, pi r](finite energy case), and 2)|F(omega)|is integrable on[-pi r, pi r]. In case 1) it is shown that the truncation error magnitude is bounded above byg(r, t) cdot sqrt{E} cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereEdenotes the signal energy andgis independent ofN_{1}, N_{2}and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above byh(r, t) cdot M cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereMis the maximum signal amplitude andhis independent ofN_{1}, N_{2}and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values ofrdose to unity.