http://ieeexplore.ieee.org TOC Alert for Publication# 18 2013 May 23 59 6 C1 C4 170 59 6 C2 C2 143 $ {varepsilon }> 0$, Guruswami and Rudra presented an $n^{O(1/ {varepsilon })}$ time algorithm to list decode appropriate folded RS codes of rate $R$ from a fraction $1-R- {varepsilon }$ of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices for a statement of the above form. Here, we give a simple linear-algebra-based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list-decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in quadratic time. We also consider a closely related family of codes, called (order $m$) derivative codes and defined over fields of large characteristic, which consist of the evaluations of $f$ as well as its first $m-1$ formal derivatives at $N$ distinct field elements. We show how our linear-algebraic methods for folded RS codes can be used to show that derivative codes can also achiev- the above optimal tradeoff. The theoretical drawback of our analysis for folded RS codes and derivative codes is that both the decoding complexity and proven worst-case list-size bound are $n^{Omega (1/ {varepsilon })}$ . By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list-size bound of $O(1/ {varepsilon }^{2})$ which is quite close to the existential $O(1/ {varepsilon })$ bound (however, the decoding complexity remains $n^{Omega (1/ {varepsilon })}$ ). Our work highlights that constructing an explicit subspace-evasive subset that has small intersection with low-dimensional subspaces—an interesting problem in pseudorandomness in its own right—could lead to explicit codes with better list-decoding guarantees.]]> 59 6 3257 3268 2769 59 6 3269 3281 3504 $z< xleq {{ n-z}over { 2}}$, where $x$ , $z$, and $n$ are dimensions of codewords, error configurations, and the ground space, respectively. We determine the capacity of nonconstant dimensional localized error correction codes when $z < {{ n}over { 3}}$.]]> 59 6 3282 3294 3670 $n$, $16leq nleq 23$, are computed, covering the four extremal odd unimodular lattices and all the 111 nonextremal unimodular lattices (both odd and even), providing thus a classification of the best wiretap lattice codes coming from unimodular lattices in dimension $n$, $8<nleq 23$. Finally, to permit lattice encoding via Construction A, the corresponding error correction codes of the best lattices are determined.]]> 59 6 3295 3303 1779 ${bf x}=(x_{1},ldots,x_{n})$ symbol-by-symbol over a communication channel. The adversarial jammer can view the transmitted symbols $x_{i}$ one at a time and can change up to a ${p}$-fraction of them. However, for each symbol $x_{i}$, the jammer's decision on whether to corrupt it or not (and on how to change it) must depend only on $x_{j}$ for $jleq i$. This is in contrast to the “classical” adversarial jammer which may base its decisions on its complete knowledge of ${bf x}$. More generally, for a delay parameter ${delta}in (0,1)$, we study the scenario in which the jammer's decision on the corruption of $x_{i}$ must depend solely on $x_{j}$ for $jleq i-{delta}{n}$. In this study, the transmitted symbols are assumed to be over a sufficiently large field $BBF$ . The sender and receiver do not share resources such as common randomness (though the sender is allowed to use stochastic encoding). We present a tight characterization of the amount - f information one can transmit in both the 0-delay and, more generally, the ${delta}$-delay online setting. We show that for 0-delay adversaries, the achievable rate asymptotically equals that of the classical adversarial model. For positive values of ${delta}$ , we consider two types of jamming: additive and overwrite. We also extend our results to a jam-or-listen online model, where the online adversary can either jam a symbol or eavesdrop on it. We present computationally efficient achievability schemes even against computationally unrestricted jammers.]]> 59 6 3304 3316 2865 59 6 3317 3331 3487 59 6 3332 3346 3916 59 6 3347 3353 910 59 6 3354 3377 5721 $C$ be a binary self-dual code with an automorphism $g$ of order $2p$, where $p$ is an odd prime, such that $g^{p}$ is a fixed point free involution. If $C$ is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code $C$ with coding theoretical ones of the subcode $C(g^{p})$ which consists of the set of fixed points of $g^{p}$, we prove that $C$ is a projective $ {BBF }_{2}langle g rangle $-module if and only if a natural projection of $C(g^{p})$ is a self-dual code. We then discuss easy-to-handle criteria to decide if $C$ is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.]]> 59 6 3378 3383 1644 $ BBZ _{4}$-linear code whose Gray image has the parameters $(72,2^{36},16)$.]]> 59 6 3384 3386 622 $n$ variables ($n$ even) of algebraic degree ranging from 2 to ${{n} over {2}}$, which implies that the maximum algebraic degree of an $n$-variable bent-negabent function is equal to ${{n} over {2}}$. Thus, we answer two open problems proposed by Parker and Pott and by Stănică ]]> 59 6 3387 3395 3163 $ {varepsilon }= k/N$ and $delta = n/N$ denote the generalized sparsity and undersampling fractions for sampling the $k$-generalized-sparse $N$ -vector $x_{0}$ according to $y=Ax_{0}$ . Here, $A$ is an $ntimes N$ measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve $ delta = delta ({varepsilon })$ separating successful from unsuccessful reconstruction of $x_{0}$ by AMP is given by $delta = M({v- repsilon }vert {rm Denoiser})$ where $M({varepsilon }vert {rm Denoiser})$ denotes the per-coordinate minimax mean squared error (MSE) of the specified, optimally tuned denoiser in the directly observed problem $y = x + z$. In short, the phase transition of a noiseless undersampling problem is identical to the minimax MSE in a denoising problem. We prove that this formula follows from state evolution and present numerical results validating it in a wide range of settings. The above formula generates numerous new insights, both in the scalar and in the nonscalar cases.]]> 59 6 3396 3433 9076 59 6 3434 3447 4317 ${ssr NP}$-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided ${ssr P}ne{ssr NP}$.]]> 59 6 3448 3450 489 59 6 3451 3465 3552 59 6 3466 3474 2673 59 6 3475 3496 6422 59 6 3497 3513 3981 $ alpha $-mixing processes. The established limit theorems allow us to design monitoring algorithms with the desirable level of the probability of false alarm and able to detect a change with probability approaching one.]]> 59 6 3514 3531 4018 59 6 3532 3553 6287 59 6 3554 3571 3994 59 6 3572 3581 2636 $d$ , with or without feedback of the past channel output symbols and a decoder, which sequentially constructs the source symbols using the channel output. The objective is to minimize the expected per-symbol distortion. For a fixed finite lookahead $dgeq 1$, we invoke the theory of controlled Markov chains to obtain an average cost optimality equation (ACOE), the solution of which, denoted by $D(d)$, is the minimum expected per-symbol distortion. With increasing $d$, $D(d)$ bridges the gap between causal encoding, $d=0$, where symbol-by-symbol encoding–decoding is optimal and the infinite lookahead case, $d=infty$, where Shannon Theoretic arguments show that separation is optimal. We extend the analysis to a system with finite-state decoders, with or without noise-free feedback. For a Bernoulli source and binary symmetric channel, under Hamming loss, we compute the optimal distortion for various source and channel parameters, and thus obtain computable bounds on $D(d)$. We also identify regions of source and channel parameters where symbol-by-symbol encoding–decoding is suboptimal. Finally, we demonstrate the wide applicability of our approach by applying it in additional coding scenarios, such as the cas- where the sequential decoder can take cost-constrained actions affecting the quality or availability of side information about the source.]]> 59 6 3582 3606 6865 59 6 3607 3618 3204 59 6 3619 3634 5928 59 6 3635 3652 7565 59 6 3653 3667 4651 59 6 3668 3692 7490 59 6 3693 3708 7642 59 6 3709 3719 3074 59 6 3720 3752 8418 ${bf x}=(x_{1}, dots, x_{n})$ bit-by-bit over a communication channel. The sender and the receiver do not share common randomness. The adversarial jammer can view the transmitted bits $x_{i}$ one at a time and can change up to a $p$-fraction of them. However, the decisions of the jammer must be made in a causal manner. Namely, for each bit $x_{i}$, the jammer's decision on whether to corrupt it or not must depend only on $x_{j}$ for $j leq i$. This is in contrast to the “classical” adversarial jamming situations in which the jammer has no knowledge of ${bf x}$, or knows ${bf x}$ completely. In this study, we present upper bounds (that hold under both the average and maximal probability of error criteria) on the capacity which hold for both deterministic and stochastic encoding schemes.]]> 59 6 3753 3763 2388 59 6 3764 3779 4397 This paper studies the multiple-access channel (MAC) with rate-limited feedback. The channel output is encoded into one stream of bits, which is provided causally to the two users at the channel input. An achievable rate region for this setup is derived, based on superposition of information, block Markov coding, and coding with various degrees of side information for the feedback link. The suggested region coincides with the Cover–Leung inner bound for large feedback rates. The result is then extended for cases where there is only a feedback link to one of the transmitters, and for a more general case where there are two separate feedback links to both transmitters. We compute achievable regions for the Gaussian MAC and for the binary erasure MAC. The Gaussian region is computed for the case of common rate-limited feedback, whereas the region for the binary erasure MAC is computed for one-sided feedback. It is known that for the latter, the Cover–Leung region is tight, and we obtain results that coincide with the feedback capacity region for high feedback rates.]]> 59 6 3780 3795 5250 59 6 3796 3810 4610 has approximated the capacity of this network up to an additive gap. The communication scheme achieving this capacity approximation is based on compress-and-forward, resulting in noise accumulation as the messages traverse the network. As a consequence, the approximation gap increases linearly with the network depth. This paper develops a computation alignment strategy that can approach the capacity of a class of layered, time-varying wireless relay networks up to an approximation gap that is independent of the network depth. This strategy is based on the compute-and-forward framework, which enables relays to decode deterministic functions of the transmitted messages. Alone, compute-and-forward is insufficient to approach the capacity as it incurs a penalty for approximating the wireless channel with complex-valued coefficients by a channel with integer coefficients. Here, this penalty is circumvented by carefully matching channel realizations across time slots to create integer-valued effective channels that are well suited to compute-and-forward. Unlike prior constant gap results, the approximation gap obtained in this paper also depends closely on the fading statistics, which are assumed to be i.i.d. Rayleigh.]]> 59 6 3811 3832 4911 59 6 3833 3847 3744 59 6 3848 3857 3212 59 6 3858 3874 5495 $K$ parallel paths between the source and the destination. While we assume that there is no direct source-destination path, the $K$ relaying paths can interfere with each other. The second class, termed as layered networks, is comprised of relays organized in layers, where links exist only between adjacent layers.]]> 59 6 3875 3891 2212 59 6 3892 3914 6493 59 6 3915 3921 1151 59 6 3922 3931 2307 59 6 3932 3938 1565 59 6 3939 3952 3368 $n$ -user wireless network in which $k$ users wish to share a unique message with all other users. Each of the $n$ users only has knowledge of its own contents and state information; this corresponds to a one-sided push-only scenario. The goal is to disseminate all messages efficiently, hopefully achieving an order-optimal spreading rate over unicast wireless random networks. First, we show that a random-push strategy—where a user sends its own or a received packet at random—is order-wise suboptimal in a random geometric graph: specifically, $Omegaleft(sqrt{n}right)$ times slower than optimal spreading. It is known that this gap can be closed if each user has “full” mobility, since this effectively creates a complete graph. We instead consider velocity-constrained mobility where at each time slot the user moves locally using a discrete random walk with velocity $v(n)$ that is much lower than full mobility. We propose a simple two-stage dissemination strategy that alternates between individual message flooding (“self promotion”) and random gossiping. We prove that this scheme achieves a close to optimal spreading rate (within only a logarithmic gap) as long as the velocity is at least $v(n)=omega (sqrt{log n/k})$. The key insight is that the mixing property introduced by the partial mobility helps users to spread in space within a relatively short period compared to the optimal spreading time, which macroscopically mimics message dissemination over a complete graph.]]> 59 6 3953 3970 4773 $epsilon$-converging time, i.e., the earliest time at which the deviation is $epsilon$ close to its steady value, and on the mean and mean-square behaviors of the displacement of node states from their initial states at large iteration number. As a special case study, performance of the broadcast-based consensus algorithm under zero-mean stochastic disturbances is analyzed, and results regarding its convergence, mean-square deviation, and mean-square displacement are given. The theoretical results presented in this study hold true regardless of the statistics of the stochastic disturbances, and are valid for arbitrary network topology as long as th- topology is connected.]]> 59 6 3971 3989 4662 $p$ and an integer $Mvert (p-1)$, the new $M$-ary sequence families of period $p$ are obtained from irreducible quadratic polynomials and known power residue-based sequence families. The new sequence families increase family sizes of the known power residue-based sequence families, but keep the maximum correlation unchanged. In the second construction, the sequences derived from the Weil representation are generalized, where each new sequence is the elementwise product of a modulated Sidel'nikov sequence and a modulated trace sequence. For positive integers $d<p$ and $Mvert (p^{n}-1)$, the new family consists of $(M-1)p^{nd}$ sequences with period $p^{n}-1$, alphabet size $Mp$, and the maximum correlation bounded by $(d+1)sqrt{p^{n}}+3$.]]> 59 6 3990 3998 2858 $(N,w,lambda_{rm a},lambda_{rm c})$-OOC with size $L$ is a family of $L;{0,1}$-sequences with length $N$, weight $w$, maximum autocorrelation $lambda_{rm a}$, and maximum cross correlation $lambda_{rm c}$. In this paper, we present two new constructions for OOCs with $lambda_{rm a}=lambda_{rm c}=1$ which are asymptotically optimal with respect to the Johnson bound. We first construct an asymptotically optimal $left(Mp^{n},M,1,1right)$ -OOC with size $(p^{n}-1)/M$ by using the structure of ${BBZ}_{p^{n}}$, the ring of integers modulo $p^{n}$, where $p$ is an odd prime with $Mvert p-1$, and $n$ is a positive integer. We then present another asymptotically optimal $(Mp_{1}cdots p_{k}, M, 1,1)$-OOC with size $(p_{1}cdots p_{k}-1)/M$ from a prod- ct of $k$ finite fields, where $p_{i}$ is an odd prime and $M$ is a positive integer such that $Mvert ,p_{i}-1$ for $1leq ileq k$. In particular, it is optimal in the case that $k=1$ and $(M-1)^{2} > p_{1}-1$.]]> 59 6 3999 4005 1975 $n$ be a positive integer. An NFSR of $n$ stages is called irreducible if the family of output sequences of any NFSR of stages less than $n$ is not included in that of the NFSR. In this paper, we prove that the density of the irreducible NFSRs of $n$ stages is larger than 0.39. This implies that it is expected to find an irreducible NFSR of $n$ stages among three randomly chosen NFSRs of $n$ stages.]]> 59 6 4006 4012 2183 ${bf Z}/(M)$ modulo 2, where $M$ is an odd integer that is composite and square-free, and ${bf Z}/(M)$ is the integer residue ring modulo $M$. A new sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial $fleft(xright)$ over ${bf Z}/(M)$ are pairwise distinct modulo 2. Such result improves a recent result obtained in our previous paper, and consequently, the set of primitive sequences over ${bf Z}/(M)$ that can be proven to be distinct modulo 2 is greatly enlarged.]]> 59 6 4013 4019 1885 59 6 4020 4024 1026 59 6 4025 4032 1662 $r$ subgroup of an elliptic curve defined over $BBF _{q}$ with embedding degree $k$. The final exponentiation in the Tate pairing is an exponentiation of an element in $BBF _{q^{k}}$ by $(q^{k}-1)/r$ . The hardest part of this computation is to raise to the power $lambda :=Phi _{k}(q)/r$, where $Phi _{k}(cdot)$ denotes the $k$th cyclotomic polynomial. Write it as $lambda =lambda _{0}+lambda _{1}q+cdots +lambda _{varphi (k)-1}q^{varphi (k)-1}$ in the $q$ -ary representation. The final exponentiation cost mostly depends on $kappa (lambda)$, the size of the maximum of $vert lambda _{i}vert $. In many parameterized pairing-friendly curves, the value $kappa $ is about $left (1- {{ 1}over { rho varphi (k)}}right)log _{2} q$ where $rho =log _{2} q/log _{2} r$, while random curves wil- have $kappa approx log _{2} q$. We investigate how this small $kappa $ is obtained for parameterized pairing-friendly elliptic curves, and show that $left (1- {{ 1}over { rho varphi (k)}}right)log _{2} q$ is the lower bound for all known construction methods of parameterized pairing-friendly curves. In the second part of our paper, we propose a method to obtain a modified Tate pairing with small $kappa $ for any pairing-friendly elliptic curves including those not belonging to parameterized families. More precisely, our method finds an integer $m$ using the lattice basis reduction such that $kappa (mlambda)=left (1- {{ 1}over { rho varphi (k)}}right)log _{2} q$. Using this modified Tate pairing, we can reduce the number of squarings in the final exponentiation by a factor of $left (1- {{ 1}over { rho varphi (k)}}right)$ from the usual Tate pairing. We apply our method to several known pairing-friendly curves to verify the expected speedup.]]> 59 6 4033 4041 2991 59 6 4042 4054 3200 59 6 4055 4056 366 59 6 C3 C3 81