Universal Lossless Compression of Erased Symbols

A source ${mmb X}$ goes through an erasure channel whose output is ${mmb Z}$. The goal is to compress losslessly ${mmb X}$ when the compressor knows ${mmb X}$ and ${mmb Z}$ and the decompressor knows ${mmb Z}$. We propose a universal algorithm based on context-tree weighting (CTW), parameterized by a memory-length parameter $ell$. We show that if the erasure channel is stationary and memoryless, and ${mmb X}$ is stationary and ergodic, then the proposed algorithm achieves a compression rate of $H(X_0vert X_{-ell}^{-1}, Z^ell)$ bits per erasure.      

High-Temperature High-Power Operation of GaInNAs Laser Diodes in the 1220–1240-nm Wavelength Range

We report on the high-temperature performance of high-power GaInNAs broad area laser diodes with different waveguide designs emitting in the 1220–1240-nm wavelength range. Large optical cavity laser structures enable a maximum continuous-wave output power of $>$8.9 W at ${T}=20 ^{circ}$C with emission at 1220 nm and are characterized by low internal losses of 0.5 cm$^{-1}$ compared to 2.9 cm$^{-1}$ for the conventional waveguide structures. High-power operation up to temperatures of 120 $^{circ}$C is observed with output powers of $>$4 W at ${T}=90 ^{circ}$C. This laser diode showed characteristic temperatures of ${T}_{0} =112$ K and ${T}_{1}=378$ K.      

On the Values of Kloosterman Sums

Given a prime $p$ and a positive integer $n$ , we show that the shifted Kloosterman sums $$sum _{x in BBF _{p^{n}}} psi (x + ax^{p^{n}-2}) = sum _{xin BBF _{p^{n}}^{ast }} psi(x + ax^{-1}) + 1, quad a inBBF _{p^{n}}^{ast }$$ where $psi$ is a nontrivial additive character of a finite field $BBF _{p^{n}}$ of $p^{n}$ elements, do not vanish if $a$ belongs to a small subfield $BBF_{p^{m}} subseteq BBF _{p^{n}}$. This complements recent results of P. Charpin and G. Gong which in turn were motivated by some applications to bent functions.      

Properties and Applications of Preimage Distributions of Perfect Nonlinear Functions

The preimage distributions of perfect nonlinear functions from an Abelian group of order $n$ to an Abelian group of order $3$ or $4$, respectively, are studied. Based on the properties of the preimage distributions of perfect nonlinear functions from an Abelian group of order $3^{r}$ to an Abelian group of order $3$, the weight distributions of the ternary linear codes $C_{Pi}$ from the perfect nonlinear functions $Pi (x)$ from $F_{3^{r}}$ to itself are determined. These results suggest that two open problems, proposed by Carlet, Ding, and Yuan in 2005 and 2006, respectively, are answered.      

On the Cusick–Cheon Conjecture About Balanced Boolean Functions in the Cosets of the Binary Reed–Muller Code

In this paper, an amplification of the Cusick–Cheon conjecture on balanced Boolean functions in the cosets of the binary Reed–Muller code $RM(k,m)$ of order $k$ and length $2^m$, in the cases where $k = 1$ or $k geq (m-1)/2$, is proved.      

Sharp $p$ -Divisibility of Weights in Abelian Codes Over ${BBZ}/p^d{BBZ}$

A theorem of McEliece on the $p$-divisibility of Hamming weights in cyclic codes over ${BBF}_p$ is generalized to Abelian codes over ${{{BBZ}/p^d{BBZ}}}$. This work improves upon results of Helleseth–Kumar–Moreno–Shanbhag, Calderbank–Li–Poonen, Wilson, and Katz. These previous attempts are not sharp in general, i.e., do not report the full extent of the $p$ -divisibility except in special cases, nor do they give accounts of the precise circumstances under which they do provide best possible results. This paper provides sharp results on $p$-divisibilities of Hamming weights and counts of any particular symbol for an arbitrary Abelian code over ${{{BBZ}/p^d{BBZ}}}$. It also presents sharp results on $2$-divisibilities of Lee and Euclidean weights for Abelian codes over ${{{BBZ}/4{BBZ}}}$.      

Logarithmic Sobolev Inequalities for Information Measures

For $alphageq 1$, the new Vajda-type information measure ${bf J}_{alpha}(X)$ is a quantity generalizing Fisher's information (FI), to which it is reduced for $alpha=2$ . In this paper, a corresponding generalized entropy power ${bf N}_{alpha}(X)$ is introduced, and the inequality ${bf N}_{alpha}(X) {bf J}_{alpha}(X)geq n$ is proved, which is reduced to the well-known inequality of Stam for $alpha=2$. The cases of equality are also determined. Furthermore, the Blachman–Stam inequality for the FI of convolutions is generalized for the Vajda information ${bf J}_{alpha}(X)$ and both families of results in the context of measure of information are discussed. That is, logarithmic Sobolev inequalities (LSIs) are written in terms of new more general entropy-type information measure, and therefore, new information inequalities are arisen. This generalization for special cases yields to the well known information measures and relative bounds.      

On Fast-Decodable Space–Time Block Codes

We focus on full-rate, fast-decodable space–time block codes (STBCs) for $2times2$ and $4times2$ multiple-input multiple-output (MIMO) transmission. We first derive conditions and design criteria for reduced-complexity maximum-likelihood (ML) decodable $2times2$ STBCs, and we apply them to two families of codes that were recently discovered. Next, we derive a novel reduced-complexity $4times2$ STBC, and show that it outperforms all previously known codes with certain constellations.      

A Note on Limited-Trial Chase-Like Algorithms Achieving Bounded-Distance Decoding

For the decoding of a binary linear block code of minimal Hamming distance $d$ over additive white Gaussian noise (AWGN) channels, a soft-decision decoder achieves bounded-distance (BD) decoding if its squared error-correction radius is equal to $d$. A Chase-like algorithm outputs the best (most likely) codeword in a list of candidates generated by a conventional algebraic binary decoder in a few trials. It is of interest to design Chase-like algorithms that achieve BD decoding with as least trials as possible. In this paper, we show that Chase-like algorithms can achieve BD decoding with only $O(d^{1/2+varepsilon })$ trials for any given positive number $varepsilon $.      

Cyclic Codes and Sequences From Generalized Coulter–Matthews Function

In this paper, we will study the exponential sum $sum_{xin {BBF}_q}chi(alpha x^{(p^k+1)/2}+beta x)$ that is related to the generalized Coulter–Matthews function $x^{(p^k+1)/2}$ with $k/{rm gcd}(m,k)$ odd. As applications, we obtain the following: the correlation distribution of a $p$-ary $m$-sequence and a decimated $m$-sequence of degree ${p^k+1 over 2}$;      

Single-Exclusion Number and the Stopping Redundancy of MDS Codes

For a linear block code ${cal C}$, its stopping redundancy is defined as the smallest number of check nodes in a Tanner graph for ${cal C}$, such that there exist no stopping sets of size smaller than the minimum distance of ${cal C}{bf .},$ Schwartz and Vardy conjectured that the stopping redundancy of a maximum-distance separable (MDS) code should only depend on its length and minimum distance.      

Exact Characterization of the Minimax Loss in Error Exponents of Universal Decoders

Universally achievable error exponents pertaining to certain families of channels (most notably, discrete memoryless channels (DMCs) and various ensembles of random codes, are studied by combining the competitive minimax approach, proposed by Feder and Merhav, with Chernoff bound and Gallager's techniques for the analysis of error exponents. In particular, we derive a single-letter expression for the largest, universally achievable fraction $xi$ of the optimum error exponent pertaining to the optimum maximum-likelihood (ML) decoding. Moreover, a simpler single-letter expression for a lower bound to $xi$ is presented. To demonstrate the tightness of this lower bound, we use it to show that $xi=1$, for the binary symmetric channel (BSC), when the random coding distribution is uniform over: i) all codes (of a given rate), and ii) all linear codes, in agreement with well-known results. We also show that $xi=1$ for the uniform ensemble of systematic linear codes, and for that of time-varying convolutional codes in the bit-error-rate sense. For the latter case, we also derive the corresponding universal decoder explicitly and show how it can be efficiently implemented using a slightly modified version of the Viterbi algorithm which employs two trellises.      

Weil-Serre Type Bounds for Cyclic Codes

We give a new method in order to obtain Weil-Serre type bounds on the minimum distance of arbitrary cyclic codes over ${BBF}_{p^e}$ of length coprime to $p$, where $e ge 1$ is an arbitrary integer. In an earlier paper we obtained Weil-Serre type bounds for such codes only when $e=1$ or $e=2$ using lengthy explicit factorizations, which seems hopeless to generalize. The new method avoids such explicit factorizations and it produces an effective alternative. Using our method we obtain Weil–Serre type bounds in various cases. By examples we show that our bounds perform very well against Bose–Chaudhuri–Hocquenghem (BCH) bound and they yield the exact minimum distance in some cases.      

Design of Shared Aperture Wideband Antennas Considering Band-Notch and Radiation Pattern Control

Two wideband antennas sharing a common aperture are presented. One antenna consists of an electrically loaded rectangle monopole, an electrically loaded inverse L-shape monopole, and a lowpass matching network between the two monopoles. It covers 30–600 MHz (VSWR ${≪}3$) with band-rejection characteristic in 86–110 MHz. The other one is a planar open-sleeve monopole, which covers 820–1200 MHz with VSWR${≪}2$. These two antennas are integrated in an aperture with size of only 420$,times,$ 200 mm $^{2}$. Both antennas are planar, which are promising candidates for either vehicular or airborne applications.      

Bit-Interleaved Coded Modulation in the Wideband Regime

The wideband regime of bit-interleaved coded modulation (BICM) in Gaussian channels is studied. The Taylor expansion of the coded modulation capacity for generic signal constellations at low signal-to-noise ratio (SNR) is derived and used to determine the corresponding expansion for the BICM capacity. Simple formulas for the minimum energy per bit and the wideband slope are given. BICM is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes. The minimum energy per bit using standard Gray mapping on $M$-PAM or $M^2$ -QAM is given by a simple formula and shown to approach ${-}$ 0.34 dB as $M$ increases. Using the low SNR expansion, a general tradeoff between power and bandwidth in the wideband regime is used to show how a power loss can be traded off against a bandwidth gain.      

Robust Precoder Adaptation for MIMO Links With Noisy Limited Feedback

In this paper, we propose two robust limited feedback designs for multiple-input multiple-output (MIMO) adaptation. The first scheme, namely, the combined design jointly optimizes the adaptation, CSIT (channel state information at the transmitter) feedback as well as index assignment strategies. The second scheme, namely, the decoupled design, focuses on the index assignment problem given an error-free limited feedback design. Simulation results show that the proposed framework has significant capacity gain compared to the naive design (designed assuming there is no feedback error). Furthermore, for large number of feedback bits $C_{rm fb}$, we show that under two-nearest constellation feedback channel assumption, the MIMO capacity loss (due to noisy feedback) of the proposed robust design scales like ${cal O}(P_e2^{-{{C_{rm fb}}over{t+1}}})$ for some positive integer $t$. Hence, the penalty due to noisy limited feedback in the proposed robust design approaches zero as $C_{rm fb}$ increases.      

Finite State Channels With Time-Invariant Deterministic Feedback

We consider capacity of discrete-time channels with feedback for the general case where the feedback is a time-invariant deterministic function of the output samples. Under the assumption that the channel states take values in a finite alphabet, we find a sequence of achievable rates and a sequence of upper bounds on the capacity. The achievable rates and the upper bounds are computable for any $N$, and the limits of the sequences exist. We show that when the probability of the initial state is positive for all the channel states, then the capacity is the limit of the achievable-rate sequence. We further show that when the channel is stationary, indecomposable, and has no intersymbol interference (ISI), its capacity is given by the limit of the maximum of the (normalized) directed information between the input $X^{N}$ and the output $Y^{N}$ , i.e., $$C = lim _{N rightarrow infty } {{ 1}over { N}} max I(X^{N} rightarrow Y^{N} )$$ where the maximization is taken over the causal conditioning probability $Q(x^{N}Vert z^{N-1})$ defined in this paper. The main idea for obtaining the results is to add causality into Gallager's results on finite state channels. The capacity results are used to show that the source–channel separation theorem holds for time-invariant determinist feedback, and if the state of the channel is known both at the encoder and the decoder, then feedback does not increase capacity.      

Performance Analysis and Design Criteria of BICM-ID With Signal Space Diversity for Keyhole Nakagami-$m$ Fading Channels

This paper generalizes the application bit-interleaved coded modulation with iterative decoding (BICM-ID) using signal space diversity (SSD) over keyhole Nakagami-$m$ fading channels. The tight union bound on the asymptotic error performance is first analytically derived. The near-optimal rotation matrix with respect to both the asymptotic performance and the convergence behavior is then determined. In particular, it is demonstrated that the suitable rotation matrix is the one that has 1) all entries equal in magnitude, 2) a high diversity order, and 3) a large minimum product of the ratios between squared distances to the power $m$ and log-squared distances to the power $m$ of the rotated constellation scaled by factors of signal-to-noise ratio (SNR) and the parameter $m$ . Various analytical and simulation results show that by employing SSD with a sufficiently large dimension, the error performance can closely approach that over an additive white Gaussian noise (AWGN) channel, even in the worst case of keyhole fading.