Equidistant binary arithmetic codes (Corresp.)

LetC(B)denote the binary cyclicANcode with generatorA, whereAB=2^{n} - 1. It is known thatC(B)is equidistant ifBis a prime powerp^{k}, where either2or-2is primitive moduloBprovidedpequiv 1 pmod{3}{rm if} k > 1. It is conjectured that these are the onlyBsuch thatC(B)is equidistant. We have verified this forB < 100 000. Several results are established that further limit the possibilities for counterexamples to the conjecture.     

All binary 3-error-correcting BCH codes of length2^m-ihave covering radius 5 (Corresp.)

Van der Horst and Berger have conjectured that the covering radius of the binary 3-error-correcting Bose-Chaudhuri-Hocquenghem (BCH) code of length2^{m} - l, m geq 4is 5. Their conjecture was proved earlier whenm equiv 0, 1, or 3 (mod 4). Their conjecture is proved whenm equiv 2(mod 4).     

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.     

Construction of de Bruijn sequences of minimal complexity

It is well known that the linear complexity of a de Bruijn sequenceSof length2^{n}is bounded below by2^{n- 1} + nforn geq 3. It is shown that this lower bound is attainable for alln.     

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.     

Algorithms for the generation of full-length shift- register sequences

Two algorithms are presented for the generation of full-length shift-register cycles, also referred to as de Bruijn sequences. The first algorithm generates2^{k cdot g(n,k)full cycles of length2^{n}, using3n + k cdot g(n, k)bits of storage, wherekis a free parameter in the range1 leq k leq 2^{((n-4)/2)}, andg(n, k)is of the order ofn - 2 log k. The second algorithm generates about2^{n^{2}/4}full cycles of length2^{n}, using aboutn^{2}/2bits of storage. In both algorithms, the time required to produce the next bit from the lastnbits is close ton. A possible application to the construction of stream ciphers is indicated.     

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.     

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.     

On single-sample robust detection of known signals with additive unknown-mean amplitude-bounded random interference--II: The randomized decision rule solution (Corresp.)

A randomized decision rule is derived and proved to be the saddlepoint solution of the robust detection problem for known signals in independent unknown-mean amplitude-bounded noise. The saddlepoint solutionphi^{0}uses an equaUy likely mixed strategy to chose one ofNBayesian single-threshold decision rulesphi_{i}^{0}, i = 1,cdots , Nhaving been obtained previously by the author. These decision rules are also all optimal against the maximin (least-favorable) nonrandomized noise probability densityf_{0}, wheref_{0}is a picket fence function withNpickets on its domain. Thee pair(phi^{0}, f_{0})is shown to satisfy the saddlepoint condition for probability of error, i.e.,P_{e}(phi^{0} , f) leq P_{e}(phi^{0} , f_{0}) leq P_{e}(phi, f_{0})holds for allfandphi. The decision rulephi^{0}is also shown to be an eqoaliir rule, i.e.,P_{e}(phi^{0}, f ) = P_{e}(phi^{0},f_{0}), for allf, with4^{-1} leq P_{e}(phi^{0},f_{0})=2^{-1}(1-N^{-1})leq2^{-1} , N geq 2. Thus nature can force the communicator to use an {em optimal} randomized decision rule that generates a large probability of error and does not improve when less pernicious conditions prevail.     

On the survival of sequence information in filters (Corresp.)

Given a binary data streamA = {a_i}_{i=o}^inftyand a filterFwhose output at timenisf_n = sum_{i=0}^{n} a_i beta^{n-i}for some complexbeta neq 0, there are at most2^{n +1)distinct values off_n. These values are the sums of the subsets of{1,beta,beta^2,cdots,beta^n}. It is shown that all2^{n+1}sums are distinct unlessbetais a unit in the ring of algebraic integers that satisfies a polynomial equation with coefficients restricted to +1, -1, and 0. Thus the size of the state space{f_n}is2^{n+1}ifbetais transcendental, ifbeta neq pm 1is rational, and ifbetais irrational algebraic but not a unit of the type mentioned. For the exceptional values ofbeta, it appears that the size of the state space{f_n}grows only as a polynomial innifmidbetamid = 1, but as an exponentialalpha^nwith1 < alpha < 2ifmidbetamid neq 1.     

A new class of codes meeting the Griesmer bound

An infinite sequence ofk-dimensional binary linear block codes is constructed with parametersn=2^{k}+2^{k-2}-15,d=2^{k-1}+2^{k-3}-8,k geq 7. Fork geq 8these codes are unique, while there are five nonisomorphic codes fork=7. By shortening these codes in an appropriate way, one finds codes meeting the Griesmer bound for2^{k-1}+2^{k-3}-15 leq d leq 2^{k-1}+2^{k-3}-8; k geq 7.     

A fundamental inequality between the probabilities of binary subgroups and cosets

The probability of a set of binaryn-tuples is defined to be the sum of the probabilities of the individualn-tuples when each digit is chosen independently with the same probabilitypof being a "one." It is shown that, under such a definition, the ratio between the probability of a subgroup of order2^{k}and any of its proper cosets is always greater than or equal to a functionF_{k}(p), whereF_{k}(p) geq 1forp leq frac{1}{2}with equality when and only whenp = frac{1}{2}. It is further shown thatF_{k}(p)is the greatest lower bound on this ratio, since a subgroup and proper coset of order2^{k}can always be found such that the ratio between their probabilities is exactlyF_{k}(p). It is then demonstrated that for a linear code on a binary symmetric channel the "tall-zero" syndrome is more probable than any other syndrome. This result is applied to the problem of error propagation in convolutional codes.     

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}.     

Sequences achieving the boundary of the entropy region for a two-source are virtually memoryless (Corresp.)

For a joint distribution{rm dist}(X,Y), the functionT(t)=min { H(Y|U): I(U wedge Y|X)=O, H(X|U)geq t}is an important characteristic. It equals the asymptotic minimum of(1/n)H(Y^{n})for random pairs of sequences(X^{n}, Y^{n}), wherefrac{1}{n} sum ^{n}_{i=1}{rm dist} X_{i} sim {rm dist} X, {rm dist} Y^{n}|X^{n} = ({rm dist} Y|X)^{n}, frac{1}{n}H(X^{n})geq t.We show that if, for(X^{n}, Y^{n})as given, the rate pair[(1/n)H(X^{n}),(1/n)H(Y^{n})]approaches the nonlinear part of the curve(t,T(t)), then the sequenceX^{n}is virtually memoryless. Using this, we determine some extremal sections of the rate region of entropy characterization problems and find a nontrivial invariant for weak asymptotic isomorphy of discrete memoryless correlated sources.     

An efficient solution of the congruencex^2 + ky^2 = mpmod{n}

The equation of the title arose in the proposed signature scheme of Ong-Schnorr-Shamir. The large integersn, kandmare given and we are asked to find any solutionx, y. It was believed that this task was of similar difficulty to that of factoring the modulusn;we show that, on the contrary, a solution can easily be found ifkandmare relatively prime ton. Under the assumption of the generalized Riemann hypothesis, a solution can be found by a probabilistic algorithm inO(log n)^{2}|loglog|k||)arithmetical steps onO(log n)-bit integers. The algorithm can be extended to solve the equationX^{2} + KY^{2} = M pmod{n}for quadratic integersK, M in {bf Z}[sqrt{d}]and to solve in integers the equationx^{3} + ky_{3} + k^{2}z^{3} - 3kxyz = m pmod{n}.     

Stability criterion for Gunn diodes with injection-limiting cathodes

According to McCumber [1] a Gunn diode with an ohmic cathode (i.e., "differential cathode conductivity"sigma_{c} = delta) is stable in a constant-voltage circuit ifn_{0}L le (n_{0}L)_{crit} equiv 2.7 times 10^{11}cm^-2wheren_{0}Lis the doping-length product. We show that the same stability criterion applies to Gunn diodes with an injection-limiting cathode(sigma_{c} rightarrow 0), if(n_{0}L)_{crit}is allowed to be a function ofsigma_{c}L. The value of(n_{0}L)_{crit}increases by 30 percent if(sigma_{c}L)varies from infinity (ohmic cathode) to zero (injection-limiting cathode). If a cathode with negative differential conductivity is realizable, it may be possible to extend the(n_{0}L)region of stable operation of Gunn diodes drastically.     

Writing on dirty paper (Corresp.)

A channel with outputY = X + S + Zis examined, The stateS sim N(0, QI)and the noiseZ sim N(0, NI)are multivariate Gaussian random variables (Iis the identity matrix.). The inputX in R^{n}satisfies the power constraint(l/n) sum_{i=1}^{n}X_{i}^{2} leq P. IfSis unknown to both transmitter and receiver then the capacity isfrac{1}{2} ln (1 + P/( N + Q))nats per channel use. However, if the stateSis known to the encoder, the capacity is shown to beC^{ast} =frac{1}{2} ln (1 + P/N), independent ofQ. This is also the capacity of a standard Gaussian channel with signal-to-noise power ratioP/N. Therefore, the stateSdoes not affect the capacity of the channel, even thoughSis unknown to the receiver. It is shown that the optimal transmitter adapts its signal to the stateSrather than attempting to cancel it.     

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 MDS extensions of generalized Reed- Solomon codes

An(n, k, d)linear code overF=GF(q)is said to be {em maximum distance separable} (MDS) ifd = n - k + 1. It is shown that an(n, k, n - k + 1)generalized Reed-Solomon code such that2leq k leq n - lfloor (q - 1)/2 rfloor (k neq 3 {rm if} qis even) can be extended by one digit while preserving the MDS property if and only if the resulting extended code is also a generalized Reed-Solomon code. It follows that a generalized Reed-Solomon code withkin the above range can be {em uniquely} extended to a maximal MDS code of lengthq + 1, and that generalized Reed-Solomon codes of lengthq + 1and dimension2leq k leq lfloor q/2 rfloor + 2 (k neq 3 {rm if} qis even) do not have MDS extensions. Hence, in cases where the(q + 1, k)MDS code is essentially unique,(n, k)MDS codes withn > q + 1do not exist.     

An algorithm for maximizing expected log investment return

Let the random (stock market) vectorX geq 0be drawn according to a known distribution functionF(x), x in R^{m}. A log-optimal portfoliob^{ast}is any portfoliobachieving maximal expectedlogreturnW^{ast}=sup_{b} E ln b^{t}X, where the supremum is over the simplexb geq 0, sum_{i=1}^{m} b_{i} = 1. An algorithm is presented for findingb^{ast}. The algorithm consists of replacing the portfoliobby the expected portfoliob^{'}, b_{i}^{'} = E(b_{i}X_{i}/b^{t}X), corresponding to the expected proportion of holdings in each stock after one market period. The improvement inW(b)after each iteration is lower-bounded by the Kullback-Leibler information numberD(b^{'}|b)between the current and updated portfolios. Thus the algorithm monotonically improves the returnW. An upper bound onW^{ast}is given in terms of the current portfolio and the gradient, and the convergence of the algorithm is established.