An upper bound on the ratio of the probabilities of subgroups and cosets (Corresp.)

Through a counterexample, Redinbo's upper bound on the ratio of the probability of a group and that of its coset is shown to be incorrect. The given example is also shown to provide a new upper bound.     

Inequalities between the probability of a subspace and the probabilities of its cosets

We consider ann-dimensional vector space overGF(q)which has a probability distribution defined on it. The sum of the probabilities over a properk-dimensional subspace is compared to a sum over a coset of this subspace. The difference of these set probabilities is related to a sum of the Fourier transforms of the distribution over a subset of the domain of the transforms. We demonstrate the existence of a coset and both an upper and a lower bound on the difference associated with this coset. The bounds depend on the maximum and nonzero minimum of the transforms as defined on a special subset of the transform domain. Two examples from coding theory are presented. The first deals with aq-ary symmetric channel while the second is concerned with a binary compound channel.     

A binary analog to the entropy-power inequality

Let {X_n}, {Y_n} be independent stationary binary random sequences with entropy H( X), H(Y), respectively. Let h(ζ)=-ζlogζ-(1-ζ)log(1-ζ), 0⩽ζ⩽1/2, be the binary entropy function and let σ(X)=h^-1 (H(X)), σ(Y)=h^-1 (H(Y)). Let z_n=X_n⊕Y_n , where ⊕ denotes modulo-2 addition. The following analog of the entropy-power inequality provides a lower bound on H(Z ), the entropy of {Z_n}: σ(Z)⩾σ(X)*σ(Y), where σ(Z)=h^-1 (H(Z)), and α*β=α(1-β)+β(1-α). When {Y _n} are independent identically distributed, this reduces to Mrs. Gerber's Lemma from A.D. Wyner and J. Ziv (1973)     

Weight distributions of cosets of two-error-correcting binary BCHcodes, extended or not

Let B be the binary two-error-correcting BCH code of length 2^m-1 and let Bˆ be the extended code of B. We give formal expressions of weight distributions of the cosets of the codes Bˆ only depending on m. We can then deduce the weight distributions of the cosets of B. When m is odd, it is well known that there are four distinct weight distributions for the cosets of B. So our main result is about the even case. Camion, Courteau, and Montpetit (see ibid., vol.38, no.7, p.1353, 1992) observe that for the lengths 15, 63, and 255 there are eight distinct weight distributions. We prove that this property holds for the codes Bˆ and B for all even m     

Property of cosets and its application to the calculation of moments of binary sequences having multipliers

A property of the sum of the elements of a coset is established and applied, with a proof of Barker's hypothesis on the properties of m sequences, to determine the first moment of a binary sequence of + 1s and ? 1s with a multiplier.     

Bit probabilities of optimal binary source codes

The probabilities of the bits produced by an optimal binary source encoder for a memoryless source are analyzed. The class of sources for which the probability of zero must equal 1/2 for any such encoder is described. Procedures that minimize (maximize) the absolute difference of the bit probabilities are discussed. Corresponding upper bounds are derived and compared for the class of uniform sources     

Undetected error probabilities of binary primitive BCH codes forboth error correction and detection

We investigate the undetected error probabilities for bounded-distance decoding of binary primitive BCH codes when they are used for both error correction and detection on a binary symmetric channel. We show that the undetected error probability of binary linear codes can be simplified and quantified if the weight distribution of the code is binomial-like. We obtain bounds on the undetected error probability of binary primitive BCH codes by applying the result to the code and show that the bounds are quantified by the deviation factor of the true weight distribution from the binomial-like weight distribution     

Salt and pepper noise removal in binary images using image block prior probabilities

During scanning and transmission, images can be corrupted by salt and pepper noise, which negatively affects the quality of subsequent graphic vectorization or text recognition. In this paper, we present a new algorithm for salt and pepper noise suppression in binary images. The algorithm consists of the computation of block prior probabilities from training noise-free images; noise level estimation; and the maximum a posteriori probability estimation of each image block. Our experiments show that the proposed method performs significantly better than the state of the art techniques.     

Error probabilities for Rician fading multichannel reception of binary andn-ary signals

Performance characteristics are derived for two different forms of multireceivers (the coherent and noncoherent) which are used with binary andN-ary signaling through the Rician fading multichannel. The coherent multireceiver is capable of perfectly measuring the channel amplitudes and phases whereas, at the other extreme, the noncoherent multireceiver implies a receiver which requires no channel measurement whatsoever. The multichannel model presumes that each transmission mode supports a specular or fixed component and a random or scatter component which fades according to the Rayleigh distribution. Heretofore, performance analyses of multichannel links have assumed that the fading obeys the Rayleigh law. This multichannel model is sufficiently general to include four practical types: the Rician and the Rayleigh fading multichannels, multichannels whose propagation modes do not fade, and those which simultaneously contain Rician and Rayleigh fading propagation paths or the so called mixed-mode multichannel. Error probabilities are graphically illustrated and compared for various multichannel models. It is found that the effectiveness of multichannel reception is highly dependent on the strength of the specular channel component relative to the mean squared value of the random channel component. In particular, multichannel reception is more effective when applied to the completely random multichannel. For special cases the error-rate expressions reduce to well-known results.     

A simple proof of the entropy-power inequality

This correspondence gives a simple proof of Shannon's entropy-power inequality (EPI) using the relationship between mutual information and minimum mean-square error (MMSE) in Gaussian channels.     

On cosets of the generalized first-order reed-muller code with low PMEPR

Golay sequences are well suited for use as codewords in orthogonal frequency-division multiplexing (OFDM), since their peak-to-mean envelope power ratio (PMEPR) in q-ary phase-shift keying (PSK) modulation is at most 2. It is known that a family of polyphase Golay sequences of length 2^m organizes in m!/2 cosets of a q-ary generalization of the first-order Reed-Muller code, RM_q(1,m). In this paper, a more general construction technique for cosets of RM _q(1,m) with low PMEPR is established. These cosets contain so-called near-complementary sequences. The application of this theory is then illustrated by providing some construction examples. First, it is shown that the m!/2 cosets of RM_q(1,m) comprised of Golay sequences just arise as a special case. Second, further families of cosets of RM_q(1,m) with maximum PMEPR between 2 and 4 are presented, showing that some previously unexplained phenomena can now be understood within a unified framework. A lower bound on the PMEPR of cosets of RM_q(1,m) is proved as well, and it is demonstrated that the upper bound on the PMEPR is tight in many cases. Finally, it is shown that all upper bounds on the PMEPR of cosets of RM_q(1,m) also hold for the peak-to-average power ratio (PAPR) under the Walsh-Hadamard transform (WHT)     

A comparison between the performances of CDMA multiuser detectors for binary image transmission

This article presents the results obtained by using CDMA multiuser detectors structures when binary images are transmitted. The performances are compared in terms of bit error rate as well as peak signal-to-noise ratio as functions of the channel signal-to-noise ratio and for different transmission scenarios. In the final section, several interesting conclusions are highlighted.     

A binary wavelet decomposition of binary images

We construct a theory of binary wavelet decompositions of finite binary images. The new binary wavelet transform uses simple module-2 operations. It shares many of the important characteristics of the real wavelet transform. In particular, it yields an output similar to the thresholded output of a real wavelet transform operating on the underlying binary image. We begin by introducing a new binary field transform to use as an alternative to the discrete Fourier transform over GF(2). The corresponding concept of sequence spectra over GF(2) is defined. Using this transform, a theory of binary wavelets is developed in terms of two-band perfect reconstruction filter banks in GF(2). By generalizing the corresponding real field constraints of bandwidth, vanishing moments, and spectral content in the filters, we construct a perfect reconstruction wavelet decomposition. We also demonstrate the potential use of the binary wavelet decomposition in lossless image coding.     

On some cosets of the first-order Reed-Muller code with highminimum weight

We study a family of particular cosets of the first-order Reed-Muller code R(1,m): those generated by special codewords, the idempotents. Thus we obtain new maximal weight distributions of cosets of R(1,7) and 84 distinct almost maximal weight distributions of cosets of R(1,9), that is, with minimum weight 240. This leads to crypotographic applications in the context of stream ciphers     

Fast conversion between binary and residue numbers

New, efficient hardware implementations are considered which perform code conversions between the three moduli (2ⁿ-1, 2 ⁿ, 2ⁿ+1) residue number systems and their binary representations. Significant hardware saving together with high-speed throughput is achieved by using only n and (n+1)-bit binary adders     

Relation between bit and symbol error probabilities for FSK withLDI detection

In M-ary, Gray coded digital communication the bit error probability is usually approximated by the symbol error probability divided by the number of bits in a symbol. This approximation is known to be excellent for phase shift keying with coherent detection at high-signal-to-noise ratios and Gaussian channel. Here it is illustrated that this approximation may also be good for M-ary frequency shift keying with limiter-discriminator-integrator detection on Gaussian, Rayleigh, and Rician channels     

On cryptographic properties of the cosets of R(1, m)

We introduce a new approach for the study of weight distributions of cosets of the Reed-Muller code of order 1. Our approach is based on the method introduced by Kasami (1968), using Pless (1963) identities. By interpreting some equations, we obtain a necessary condition for a coset to have a “high” minimum weight. Most notably, we are able to distinguish such cosets which have three weights only. We then apply our results to the problem of the nonlinearity of Boolean functions. We particularly study the links between this criterion and the propagation characteristics of a function     

Comparison between the Jacoby-Kost and Horiguchi-Morita binary two-thirds rate modulation codes

Bit error rate graphs obtained by simulating both Viterbi decoding and sliding block decoding on the binary symmetric channel and a two-state bursty channel are presented for the R = 2/3, (d,k) = (1,7) codes of Jacoby-Kost and Horiguchi-Morita. The measured power spectral densities of both codes are compared to that of the source, generating all (d,k) = (1,7) sequences.     

A simple derivation of the undetected error probabilities of complementary codes

Recently, Fu, Klove, and Wei (2003) have shown that the undetected error probability of a binary code is related to that of its complement, and the undetected error probability of a constant-weight binary code is related to that of its complement relative to the set of all constant-weight vectors. We generalize these relations to cover the complements of any binary or nonbinary code relative to a distance-invariant code containing the first code. We prove the generalization using a much simpler argument than the published proofs of the special cases.