共查询到20条相似文献,搜索用时 15 毫秒
1.
《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
2.
《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
3.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(3):480-484
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, wherek is 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}/2 bits of storage. In both algorithms, the time required to produce the next bit from the lastn bits is close ton . A possible application to the construction of stream ciphers is indicated. 相似文献
4.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1969,15(3):408-413
It is shown that ifm neq 8, 12 andm > 6 , there are some binary primitive BCH codes (BCH codes in a narrow sense) of length2^{m} - 1 whose minimum weight is greater than the BCH bound. This gives a negative answer to the question posed by Peterson [1] of whether or not the BCH bound is always the actual minimum weight of a binary primitive BCH code. It is also shown that for any evenm geq 6 , there are some binary cyclic codes of length2^{m} - 1 that have more information digits than the primitive BCH codes of length2^{m} - 1 with the same minimum weight. 相似文献
5.
《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
6.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1978,24(5):627-628
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} . 相似文献
7.
《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
8.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1976,22(3):348-349
The coset leader of greatest weight in the 3-error-correcting BCH code of length2^{m}-1 has weight 5, for oddm geq 5 . 相似文献
9.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1976,22(6):737-745
A particular shortening technique is applied to majority logic decodable codes of length2^{t} . The shortening technique yields new efficient codes of lengthsn = 2^{p} , wherep is a prime, e.g., a (128,70) code withd_{maj} = 16 . For moderately long code lengths (e.g.,n = 2^{11} or 2^{13}) , a 20-25 percent increase in efficiency can be achieved over the best previously known majority logic decodable codes. The new technique also yields some efficient codes for lengthsn = 2^{m} wherem is a composite number, for example, a (512,316) code withd_{maj} = 32 code which has 42 more information bits than the previously most efficient majority logic decodable code. 相似文献
10.
《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
11.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1978,24(2):257-258
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 4 is 5. Their conjecture was proved earlier whenm equiv 0, 1 , or 3 (mod 4). Their conjecture is proved whenm equiv 2 (mod 4). 相似文献
12.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1977,23(6):776-778
It is shown that a high-radix fast Fourier transform (FFT) with generatorgamma = 3 over GF(F_{n}) , whereF_{n} = 2^{2}^{n'} + 1 is 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}) . 相似文献
13.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1983,29(3):448-450
We present an efficient maximum likelihood decoding algorithm for the punctured binary Reed-Muller code of order(m - 3) and length2^{m} - 1, M geq 3 , and we give formulas for the weight distribution of coset leaders of such codes. 相似文献
14.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1982,28(4):665-668
The modular distance induces a metric if and only if the nonadjacent form of the modulusM has 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 5 andj-igeq 2; 3) 2^{n} pm 2^{j} , wheren -j geq 2; 4) 2^{n} . 相似文献
15.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1983,29(4):597-600
We construct0, pm 1 sequences of length(q^{2l+1}-l)/(q-1) , whereq=2^{s} , with out-of-phase periodic autocorrelation0 , and in-phase correlationq^{2l}; SUch that the peak factor of radiation is(q^{2l+1}-1) /(q^{2+l}-q^{2l}) , which is close to1 asq becomes large. 相似文献
16.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1971,17(4):368-371
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} wherem denotes 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 messagem and 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. 相似文献
17.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1981,27(5):548-555
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 8 these 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 . 相似文献
18.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1986,32(3):423-426
Nondegenerate quadrics of PG(2l, 2^{s}) have been used to construct ternary sequences of length(2^{2sl+1} - 1)/(2^{s} - 1) with perfect autocorrelation function. The same construction can be used for degenerate quadrics for this case as well as quadrics of PG(N,q) , withN arbitrary andq = p^{s} , for any primep . This is possible because it is shown that ifQ subseteq {rm PG} (N, q) is a quadric, possibly degenerate, that has the same size as a hyperplane, then, providedQ itself is not a hyperplane, the hyperplanes of PG(N,q) intersectQ in three sizes. These sizes depend on whetherN is even or odd and the degeneracy ofQ . Finally, a connection to maximum period linear recursive sequences is made. 相似文献
19.
Construction of de Bruijn sequences of minimal complexity 总被引:1,自引:0,他引:1
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1984,30(5):705-709
It is well known that the linear complexity of a de Bruijn sequenceS of length2^{n} is bounded below by2^{n- 1} + n forn geq 3 . It is shown that this lower bound is attainable for alln . 相似文献
20.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1987,33(5):744-747
Letalpha_{n} denote the number of cosets with minimum weightn of the(2^{m}, m + 1) Reed-Muller code. Thealpha_{n} for2^{m-2} leq n < 2^{m-2} + 2^{m - 4} is determined. 相似文献