共查询到20条相似文献,搜索用时 31 毫秒
1.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1975,21(4):460-462
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. 相似文献
2.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(3):409-414
The weight enumerator of a code is the polynomial begin{equation} W(x,y)= sum_{r=0}^n A_r x^{n-r} y^r, end{equation} wheren denotes the block length andA_r , denotes the number of codewords of weightr . LetC be a self-dual code overGF(q) in which every weight is divisible byc . Then Gleason's theorem states that 1) ifq = 2 andc = 2, the weight enumerator ofC is a sum of products of the polynomialsx^2 + y^2 andx^2y^2 (x^2 - y^2 )^2 ifq = 2 andc = 4, the weight enumerator is a sum of products ofx^8 + 14x^4 y^4 + y^8 andx^4 y^4 (x^4 - y^4)^4 ; and 3) ifq = 3 andc = 3, the weight enumerator is a sum of products ofx^4 + 8xy^3 andy^3(x^3 - y^3)^3 . In this paper we give several proofs of Gleason's theorem. 相似文献
3.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(2):385-388
LetV be 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 ifnu has 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-q representation of the numberx . LetS be 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 thatS is 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 . 相似文献
4.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1970,16(6):745-751
In this paper, we establish the following result. Theorem:A_i , the number of codewords of weighti in the second-order binary Reed-Muller code of length2^m is given byA_i = 0 unlessi = 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} 相似文献
5.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1986,32(5):706-709
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} - 1 withm geq 8 and designed distance2t + 1 with4 leq t leq 5 is 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 + 1 andm geq 16 ; (2)t = 4, j geq 2t + 3 and10 leq m leq 15 ; (3)t=4, j geq 2t+5 and8 leq m leq 9 ; (4)t=5,j geq 2t+ 1 andm geq 20 ; (5)t=5, j geq 2t+ 3 and12 leq m leq 19 ; (6)t=5, j geq 2t+ 5 and10 leq m leq 11 ; (7)t=5, j geq 2t + 7 andm=9 ; (8)t= 5, j geq 2t+ 9 andm = 8 . 相似文献
6.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1973,19(6):769-772
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. LetX andY be 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 ofX andY , 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 . 相似文献
7.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(2):395-403
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 rceil denotes 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 = 2 oru_{i-1}-u_{i} geq 2 for2 leq i leq p . 相似文献
8.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(4):523-525
LetX_1,X_2,cdots be a sequence of independent identically distributed observations with a common meanmu . Assume that0 leq X_i leq 1 with probability 1. We show that for eachvarepsilon > 0 there exists an integerm , a finite-valued statisticT_n = T_n(X_1, cdots , X_n) in {t_1,cdots,t_m} and a real-valued functiond defined on{t_1,cdots,t_m} such thati )T_{n+1} = f_n(T_n,X_{n+1}) ; ii)P[lim sup mid d(T_n) - mu mid leq varepsilon] = 1 . Thus we have a recursive-like estimate ofmu , for which the data are summarized for eachn by one ofm states and which converges to withinvarepsilon ofmu with probability 1. The constraint on memory here is time varying as contrasted to the time-invariant constraint that would haveT_{n+1} = f(T_n, X_{n+1}) for alln . 相似文献
9.
Duadic Codes 总被引:3,自引:0,他引:3
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(5):709-714
A new family of binary cyclic(n,(n + 1)/2) and(n,(n - 1)/2) codes are introduced, which include quadratic residue (QR) codes whenn is prime. These codes are defined in terms of their idempotent generators, and they exist for all oddn = p_{1}^{a_{1}} p_{2}^{a_{2}} cdots p_{r}^{a_{r}} where eachp_{i} is a primeequiv pm 1 pmod{8} . Dual codes are identified. The minimum odd weight of a duadic(n,(n + 1)/2) code satisfies a square root bound. When equality holds in the sharper form of this bound, vectors of minimum weight hold a projective plane. The unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent(73,37,9) duadic codes. All duadic codes of length less than127 are identified, and the minimum weights of their extensions are given. One of the duadic codes of length113 has greater minimum weight than the QR code of that length. 相似文献
10.
Achievable rates for multiple descriptions 总被引:12,自引:0,他引:12
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1982,28(6):851-857
11.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1987,33(1):69-76
Consider separate encoding of correlated sourcesX^{n}=(X_{l}, cdots ,X_{n}), Y^{n} = (Y_{l}, cdots ,Y_{n}) for the decoder to reliably reproduce a function{F(X_{i}, Y_{i})}^{n}_{i=1} . We establish the necessary and sufficient condition for the set of all achievable rates to coincide with the Slepian-Wolf region whenever the probability densityp(x,y) is positive for all(x,y) . 相似文献
12.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1971,17(2):140-143
Letx_1, cdots ,x_n andy_1, cdots, y_n be input and output sequences of a channel. In the case of memoryless input sources, the following inequality on mutual information is well known: begin{equation} I((x_1, cdots ,x_n),(y_1, cdots ,y_n)) geq sum I(x_i, y_i). end{equation} It is straightforward to show that the inequality sign is reversed if the channel instead of the input source is memoryless. In this paper we establish these inequalities when the input and output are functions instead of sequences. 相似文献
13.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1979,25(1):105-109
LetV be a binary linear(n,k) -code defined by a check matrixH with columnsh_{1}, cdots ,h_{n} , and leth(x) = 1 ifx in {h_{1}, cdots , h_{n} , andh(x) = 0 ifx in neq {h_{1}, cdots ,h_{n}} . A combinatorial argument relates the Walsh transform ofh(x) with the weight distributionA(i) of the codeV for smalli(i< 7) . This leads to another proof of the Plessi th power moment identities fori < 7 . This relation also provides a simple method for computing the weight distributionA(i) for smalli . The implementation of this method requires at most(n-k+ 1)2^{n-k} additions and subtractions,5 .2^{n-k} multiplications, and2^{n-k} memory cells. The method may be very effective if there is an analytic expression for the characteristic Boolean functionh(x) . This situation will be illustrated by several examples. 相似文献
14.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(5):662-664
Forx(t) either a deterministic or stochastic signal band-limited to the normalized frequency intervalmidomegamid leq pi , explicit coefficients{ a_{kn} } are exhibited that have the property that begin{equation} lim_{n rightarrow infty} parallel x(t) - sum_{1}^n a_{kn} x(t - kT) parallel = 0 end{equation} in an appropriate norm and for any constant intersample spacingT satisfying0 < T < fac{1}{2} ; that is,x(t) may be approximated arbitrarily well by a linear combination of past samples taken at any constant rate that exceeds twice the associated Nyquist rate. Moreover, the approximation ofx(t) is uniform in the sense that the coefficients{ a_{kn} } do not depend on the detailed structure ofx(t) but are absolute constants for any choice ofT . The coefficients that are obtained provide a sharpening of a previous result by Wainstein and Zubakov where a rate in excess of three times the Nyquist rate was required. 相似文献
15.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1975,21(1):93-95
It is known that under certain restrictions on the posterior density and assigned cost function, the Bayes estimate of a random parameter is the conditional mean. The restrictions on the cost function are that it must be a symmetric convex upward function of the difference between the parameter and the estimate. In this correspondence, asymmetrical cost functions of the following form are examined: begin{equation} C(a, hat{a})= begin{cases} f_1(a- hat{a}),& a geq hat{a} \ f_2(hat{a}- a),& a < hat{a} end{cases} end{equation} wheref_1(cdot), f_2(cdot) are both twice-differentiable convex upward positive functions on[0, infty] that intersect the origin. It is shown that for posterior densities satisfying a certain symmetry condition, the biased Bayes estimate is a generalized median. Furthermore, for linear polynomial functionsf_1(cdot), f_2(cdot) , the unbiased Bayes estimate is shown to be the conditional mean. 相似文献
16.
17.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1972,18(2):310-312
Given a binary data streamA = {a_i}_{i=o}^infty and a filterF whose output at timen isf_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 unlessbeta is 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} ifbeta is transcendental, ifbeta neq pm 1 is rational, and ifbeta is 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 inn ifmidbetamid = 1 , but as an exponentialalpha^n with1 < alpha < 2 ifmidbetamid neq 1 . 相似文献
18.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1987,33(6):919-923
LetC be the cyclic product code ofp single parity check codes of relatively prime lengthsn_{1}, n_{2},cdots , n_{p} (n_{1} < n_{2} < cdots < n_{p}) . It is proven thatC can correct2^{P-2}+2^{p-3}-1 bursts of lengthn_{1} , andlfloor(max{p+1, min{2^{p-s}+s-1,2^{p-s}+2^{p-s-1}}}-1)/2rfloor bursts of lengthn_{1}n_{2} cdots n_{s} (2leq s leq p-2) . Forp=3 this means thatC is double-burst-n_{1} -correcting. An efficient decoding algorithm is presented for this code. 相似文献
19.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1970,16(6):717-719
A construction is given that combines an(n, M_1, d_1) code with an(n, M_2, d_2 = [frac{1}{2}(d_1 + 1)]) code to form a(2n, M_1 M_2, d_1) code. This is used to construct a new family of nongroup single-error correcting codes of all lengthsn from2^m to 3 ·2^{m-1} - 1 , for everym geq 3 . These codes have more codewords than any group code of the same length and minimum distance. A number of other nongroup codes are also obtained. Examples of the new codes are (16,2560,3) and (16,36,7) codes, both having more codewords than any comparable group code. 相似文献
20.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1966,12(2):148-153
Given a graphG ofn nodes. We wish to assign to each nodei(i = 1, 2, cdots n) a unique binary codec_{i} of lengthm such that, if we denote the Hannuing distance betweenc_{i} andc_{j} asH(c_{i}, c_{j}) , thenH(c_{i}, c_{j})leq T if nodesi andj are adjacent (i.e., connected by a single branch), andH(c_{i}, c_{j}) geq T+1 otherwise. If such a code exists, then we say thatG is doable for the value ofT and tn associated with this code. In this paper we prove various properties relevent to these codes. In particular we prove 1) that for every graphG there exists anm andT such thatG is doable, 2) for every value ofT there exists a graphG̃ which is notT doable, 3) ifG isT' doable, then it isT'+ 2p doable forp = 0, 1, 2, cdots , and is doable for allT geq 2T' ifT' is odd, and is doable for allT geq 2T' + 1 ifT' is even. In theory, the code can be synthesized by employing integer linear programming where eitherT and/orm can be minimized; however, this procedure is computationally infeasible for values ofn andm in the range of about10 or greater. 相似文献