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 ${{{BBZ}/4{BBZ}}}$.      

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.      

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.      

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.      

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 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.      

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.      

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.      

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.      

Lattices for Distributed Source Coding: Jointly Gaussian Sources and Reconstruction of a Linear Function

Consider a pair of correlated Gaussian sources $(X_1,X_2)$. Two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder. The decoder is interested in reconstructing a linear combination of $X_1$ and $X_2$ to within a mean-square distortion of $D$. We obtain an inner bound to the optimal rate–distortion region for this problem. A portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components $X_1$ and $X_2$ first. This results in a better rate region for certain parameter values. Our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory. We then consider the case of linear reconstruction of $K$ sources and provide an inner bound to the optimal rate–distortion region. Some parts of the inner bound are achieved using the following coding structure: lattice vector quantization followed by “correlated” lattice-structured binning.      

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 ${p^k+1 over 2}$;      

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.      

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.      

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.      

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.      

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 $.      

Diversity Product Properties of Lu–Kumar's Space-Time Codes

In this paper, we show that the diversity products of the full transmit diversity space-time block codes proposed by Lu–Kumar (we call them Lu–Kumar's codes) with quadratic-amplitude modulation (QAM) constellations are lower bounded by $4$. We present a sufficient condition on the minimum Hamming weight of the linear binary full-rank space-time code such that this lower bound is met. We show that the special Lu–Kumar codes does satisfy the sufficient condition, and therefore, the diversity products of the special Lu–Kumar codes are $4$, where “special” means that the linear binary full-rank space-time codes are not general but specially constructed by Lu–Kumar.      

Doubly-Generalized LDPC Codes: Stability Bound Over the BEC

The iterative decoding threshold of low-density parity-check (LDPC) codes over the binary erasure channel (BEC) fulfills an upper bound depending only on the variable and check nodes with minimum distance $2$. This bound is a consequence of the stability condition, and is here referred to as stability bound. In this paper, a stability bound over the BEC is developed for doubly-generalized LDPC codes, where variable and check nodes can be generic linear block codes, assuming maximum a posteriori erasure correction at each node. It is proved that also in this generalized context the bound depends only on the variable and check component codes with minimum distance $2$. A condition is also developed, namely, the derivative matching condition, under which the bound is achieved with equality. The stability bound leads to consider single parity-check codes used as variable nodes as an appealing option to overcome common problems created by generalized check nodes.