The rank and kernel of extended 1-perfect /spl Zopf//sub 4/-linear and additive non-/spl Zopf//sub 4/-linear codes

A binary extended 1-perfect code of length n + 1 = 2/sup t/ is additive if it is a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//. The punctured code by deleting a /spl Zopf//sub 2/ coordinate (if there is one) gives a perfect additive code. 1-perfect additive codes were completely characterized and by using that characterization we compute the possible parameters /spl alpha/, /spl beta/, rank, and dimension of the kernel for extended 1-perfect additive codes. A very special case is that of extended 1-perfect /spl Zopf//sub 4/-linear codes.     

Cyclic codes over GR(4/sup m/) which are also cyclic over /spl Zopf//sub 4/

Let GR(4/sup m/) be the Galois ring of characteristic 4 and cardinality 4/sup m/, and /spl alpha/_={/spl alpha//sub 0/,/spl alpha//sub 1/,...,/spl alpha//sub m-1/} be a basis of GR(4/sup m/) over /spl Zopf//sub 4/ when we regard GR(4/sup m/) as a free /spl Zopf//sub 4/-module of rank m. Define the map d/sub /spl alpha/_/ from GR(4/sup m/)[z]/(z/sup n/-1) into /spl Zopf//sub 4/[z]/(z/sup mn/-1) by d/spl alpha/_(a(z))=/spl Sigma//sub i=0//sup m-1//spl Sigma//sub j=0//sup n-1/a/sub ij/z/sup mj+i/ where a(z)=/spl Sigma//sub j=0//sup n-1/a/sub j/z/sup j/ and a/sub j/=/spl Sigma//sub i=0//sup m-1/a/sub ij//spl alpha//sub i/, a/sub ij//spl isin//spl Zopf//sub 4/. Then, for any linear code C of length n over GR(4/sup m/), its image d/sub /spl alpha/_/(C) is a /spl Zopf//sub 4/-linear code of length mn. In this article, for n and m being odd integers, it is determined all pairs (/spl alpha/_,C) such that d/sub /spl alpha/_/(C) is /spl Zopf//sub 4/-cyclic, where /spl alpha/_ is a basis of GR(4/sup m/) over /spl Zopf//sub 4/, and C is a cyclic code of length n over GR(4/sup m/).     

Generalized MacWilliams identities for /spl Zopf//sub 4/-linear codes

In this correspondence, the generalized MacWilliams identities between the support weight distributions of a /spl Zopf//sub 4/-linear code of type 4/sup k/ and that of its dual are established.     

Improving the Gilbert-Varshamov bound for q-ary codes

Given positive integers q,n, and d, denote by A/sub q/(n,d) the maximum size of a q-ary code of length n and minimum distance d. The famous Gilbert-Varshamov bound asserts that A/sub q/(n,d+1)/spl ges/q/sup n//V/sub q/(n,d) where V/sub q/(n,d)=/spl Sigma//sub i=0//sup d/ (/sub i//sup n/)(q-1)/sup i/ is the volume of a q-ary sphere of radius d. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant /spl alpha/ less than (q-1)/q there is a positive constant c such that for d/spl les//spl alpha/n A/sub q/(n,d+1)/spl ges/cq/sup n//V/sub q/(n,d)n. This confirms a conjecture by Jiang and Vardy.     

Asymptotic properties on codeword lengths of an optimal FV code for general sources

This correspondence is concerned with asymptotic properties on the codeword length of a fixed-to-variable length code (FV code) for a general source {X/sup n/}/sub n=1//sup /spl infin// with a finite or countably infinite alphabet. Suppose that for each n /spl ges/ 1 X/sup n/ is encoded to a binary codeword /spl phi//sub n/(X/sup n/) of length l(/spl phi//sub n/(X/sup n/)). Letting /spl epsiv//sub n/ denote the decoding error probability, we consider the following two criteria on FV codes: i) /spl epsiv//sub n/ = 0 for all n /spl ges/ 1 and ii) lim sup/sub n/spl rarr//spl infin///spl epsiv//sub n/ /spl les/ /spl epsiv/ for an arbitrarily given /spl epsiv/ /spl isin/ [0,1). Under criterion i), we show that, if X/sup n/ is encoded by an arbitrary prefix-free FV code asymptotically achieving the entropy, 1/nl(/spl phi//sub n/(X/sup n/)) - 1/nlog/sub 2/ 1/PX/sup n/(X/sup n/) /spl rarr/ 0 in probability as n /spl rarr/ /spl infin/ under a certain condition, where P/sub X//sup n/ denotes the probability distribution of X/sup n/. Under criterion ii), we first determine the minimum rate achieved by FV codes. Next, we show that 1/nl(/spl phi//sub n/(X/sup n/)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.     

Lossless and near-lossless source coding for multiple access networks

A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences {X/sub i/}/sub i=1//sup /spl infin// and {Y/sub i/}/sub i=1//sup /spl infin// is drawn independent and identically distributed (i.i.d.) according to joint probability mass function (p.m.f.) p(x,y); the encoder for each source operates without knowledge of the other source; the decoder jointly decodes the encoded bit streams from both sources. The work of Slepian and Wolf describes all rates achievable by MASCs of infinite coding dimension (n/spl rarr//spl infin/) and asymptotically negligible error probabilities (P/sub e//sup (n)//spl rarr/0). In this paper, we consider the properties of optimal instantaneous MASCs with finite coding dimension (n相似文献   

On the generators of /spl Zopf//sub 4/ cyclic codes of length 2/sup e/

Results are presented on the generators of ideals in the ring /spl Zopf//sub 4/[x]/(x/sup n/-1). In particular, each ideal (cyclic code) has a unique distinguished set of generators that characterizes any cyclic code. Some results about dual codes are also included.     

A class of 1-generator quasi-cyclic codes

If R = F/sub q/[x/spl rceil/]/(x/sup m/ - 1), S = F/sub qn/[x]/(x/sup m/ - 1), we define the mapping a_(x) /spl rarr/ A(x) =/spl sigma//sub 0//sup n-1/a/sub i/(x)/spl alpha//sub i/ from R/sup n/ onto S, where (/spl alpha//sub 0/, /spl alpha//sub i/,..., /spl alpha//sub n-1/) is a basis for F/sub qn/ over F/sub q/. This carries the q-ray 1-generator quasicyclic (QC) code R a_(x) onto the code RA(x) in S whose parity-check polynomial (p.c.p.) is defined as the monic polynomial h(x) over F/sub q/ of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of x/sub m/ 1 over F/sub q/ and F/sub qn/ are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal in S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for F/sub qn/ over F/sub q/, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes.     

Generalization of the /spl Sigma//sub 0/-rank criteria for space-time codes

Recently, Liu et al., developed the /spl Sigma//sub 0/-rank criteria for space-time codes. It provides a sufficient condition on codeword and generator matrices defined over finite rings /spl Zopf//sub 2k/(j) to ensure full spatial diversity with 2/sup 2k/-QAM modulation. Here, we generalize the /spl Sigma//sub 0/-rank criteria and derive a sufficient condition on the generator matrices defined over finite rings /spl Zopf//sub 2l/(j) to ensure full spatial diversity with 2/sup 2k/-QAM modulation for any positive integer l/spl les/k. We also show that generator matrices defined over GF(2) satisfying the BPSK stacking construction constraint of Mammons and El Gamal achieve full spatial diversity when used with 2/sup 2k/-QAM modulation.     

Channel capacity and state estimation for state-dependent Gaussian channels

We formulate a problem of state information transmission over a state-dependent channel with states known at the transmitter. In particular, we solve a problem of minimizing the mean-squared channel state estimation error E/spl par/S/sup n/ - S/spl circ//sup n//spl par/ for a state-dependent additive Gaussian channel Y/sup n/ = X/sup n/ + S/sup n/ + Z/sup n/ with an independent and identically distributed (i.i.d.) Gaussian state sequence S/sup n/ = (S/sub 1/, ..., S/sub n/) known at the transmitter and an unknown i.i.d. additive Gaussian noise Z/sup n/. We show that a simple technique of direct state amplification (i.e., X/sup n/ = /spl alpha/S/sup n/), where the transmitter uses its entire power budget to amplify the channel state, yields the minimum mean-squared state estimation error. This same channel can also be used to send additional independent information at the expense of a higher channel state estimation error. We characterize the optimal tradeoff between the rate R of the independent information that can be reliably transmitted and the mean-squared state estimation error D. We show that any optimal (R, D) tradeoff pair can be achieved via a simple power-sharing technique, whereby the transmitter power is appropriately allocated between pure information transmission and state amplification.     

Asymptotic improvement of the Gilbert-Varshamov bound on the size of binary codes

Given positive integers n and d, let A/sub 2/(n,d) denote the maximum size of a binary code of length n and minimum distance d. The well-known Gilbert-Varshamov bound asserts that A/sub 2/(n,d)/spl ges/2/sup n//V(n,d-l), where V(n,d) = /spl sigma//sub i=0//sup d/(/sub i//sup n/) is the volume of a Hamming sphere of radius d. We show that, in fact, there exists a positive constant c such that A/sub 2/(n, d)/spl ges/c2/sup n//V(n,d-1)log/sub 2/V(n, d-1) whenever d/n/spl les/0.499. The result follows by recasting the Gilbert-Varshamov bound into a graph-theoretic framework and using the fact that the corresponding graph is locally sparse. Generalizations and extensions of this result are briefly discussed.     

Characterization of accumulation layer capacitance for extracting data on high-/spl kappa/ gate dielectrics

A new parameter extraction technique has been outlined for high-/spl kappa/ gate dielectrics that directly yields values of the dielectric capacitance C/sub di/, the accumulation layer surface potential quotient, /spl beta//sub acc/, the flat-band voltage, the surface potential /spl phi//sub s/, the dielectric voltage, the channel doping density and the interface charge density at flat-band. The parallel capacitance, C/sub p/(=C/sub sc/+C/sub it/), was found to be an exponential function of /spl phi//sub s/ in the strong accumulation regime, for seven different high-/spl kappa/ gate dielectrics. The slope of the experimental lnC/sub p/(/spl phi//sub s/) plot, i.e., |/spl beta//sub acc/|, was found to depend strongly on the physical properties of the high-/spl kappa/ dielectric, i.e., was inversely proportional to [(/spl phi//sub b/m/sup *//m)/sup 1/2/K/C/sub di/], where /spl phi//sub b/ is the band offset, and m/sup */ is the effective tunneling mass. Extraction of /spl beta//sub acc/ represented an experimental carrier confinement index for the accumulation layer and an experimental gate-dielectric direct-tunneling current index. /spl beta//sub acc/ may also be an effective tool for monitoring the effects of post-deposition annealing/processing.     

Markov types and minimax redundancy for Markov sources

Redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length. In the minimax scenario, one designs the best code for the worst source within the class. Such minimax redundancy comes in two flavors: average minimax or worst case minimax. We study the worst case minimax redundancy of universal block codes for Markovian sources of any order. We prove that the maximal minimax redundancy for Markov sources of order r is asymptotically equal to 1/2m/sup r/(m-1)log/sub 2/n+log/sub 2/A/sub m//sup r/-(lnlnm/sup 1/(m-1)/)/lnm+o(1), where n is the length of a source sequence, m is the size of the alphabet, and A/sub m//sup r/ is an explicit constant (e.g., we find that for a binary alphabet m=2 and Markov of order r=1 the constant A/sub 2//sup 1/=16/spl middot/G/spl ap/14.655449504 where G is the Catalan number). Unlike previous attempts, we view the redundancy problem as an asymptotic evaluation of certain sums over a set of matrices representing Markov types. The enumeration of Markov types is accomplished by reducing it to counting Eulerian paths in a multigraph. In particular, we propose exact and asymptotic formulas for the number of strings of a given Markov type. All of these findings are obtained by analytic and combinatorial tools of analysis of algorithms.     

On the complexity of multicovering radii

The multicovering radius is a generalization of the covering radius. In this correspondence, we show that lower-bounding the m-covering radius of an arbitrary binary code is NP-complete when m is polynomial in the length of the code. Lower-bounding the m-covering radius of a linear code is /spl Sigma//sub 2//sup P/-complete when m is polynomial in the length of the code. If P is not equal to NP, then the m-covering radius of an arbitrary binary code cannot be approximated within a constant factor or within a factor n/sup /spl epsi// where n is the length of the code and /spl epsi/<1, in polynomial time. Note that the case when m=1 was also previously unknown. If NP is not equal to /spl Sigma//sub 2//sup P/,then the m-covering radius of a linear code cannot be approximated within a constant factor or within a factor n/sup /spl epsi// where n is the length of the code and /spl epsi/<1, in polynomial time.     

Order statistics from a logistic distribution and applications to survival and reliability analysis

Joint moments involving arbitrary powers of order statistics are the main concern. Consider order statistics u/sub 1/ /spl les/ u/sub 2/ /spl les/ /spl middot//spl middot//spl middot/ /spl les/ u/sub k/ coming from a simple random sample of size n from a real continuous population where u/sub 1/ = x/sub r(1):n/ is order-statistic #r/sub 1/, u/sub 2/ = x/sub r(1)+r(2):n/ is order statistic #(r/sub 1/ + r/sub 2/), et al., and u/sub k/ = x/sub r(1)+/spl middot//spl middot//spl middot/+r(k):n/ is order statistic #(r/sub 1/ +/spl middot//spl middot//spl middot/+ r/sub k/). Product moments are examined of the type E[u/sub 1//sup /spl alpha/(1)/ /spl middot/ u/sub 2//sup /spl alpha/(2)//sub /spl middot/ /spl middot//spl middot//spl middot//spl middot//u/sub k//sup /spl alpha/(k)/] where /spl alpha//sub 1/, ..., /spl alpha//sub k/ are arbitrary quantities that might be complex numbers, and E[/spl middot/] denotes the s-expected value. Some explicit evaluations are considered for a logistic population. Detailed evaluations of all integer moments of u/sub 1/ and recurrence relations, recurring only on the order of the moments, are given. Connections to survival functions in survival analysis, hazard functions in reliability situations, real type-1, type-2 /spl beta/ and Dirichlet distributions are also examined. Arbitrary product moments for the survival functions are evaluated. Very general results are obtained which can be used in many problems in various areas.     

Entropy and the law of small numbers

Two new information-theoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary information-theoretic techniques it is shown that, when S/sub n/=/spl Sigma//sub i=1//sup n/X/sub i/ is the sum of the (possibly dependent) binary random variables X/sub 1/,X/sub 2/,...,X/sub n/, with E(X/sub i/)=p/sub i/ and E(S/sub n/)=/spl lambda/, then D(P(S/sub n/)/spl par/Po(/spl lambda/)) /spl les//spl Sigma//sub i=1//sup n/p/sub i//sup 2/+[/spl Sigma//sub i=1//sup n/H(X/sub i/)-H(X/sub 1/,X/sub 2/,...,X/sub n/)] where D(P(S/sub n/)/spl par/Po(/spl lambda/)) is the relative entropy between the distribution of S/sub n/ and the Poisson (/spl lambda/) distribution. The first term in this bound measures the individual smallness of the X/sub i/ and the second term measures their dependence. A general method is outlined for obtaining corresponding bounds when approximating the distribution of a sum of general discrete random variables by an infinitely divisible distribution. Second, in the particular case when the X/sub i/ are independent, the following sharper bound is established: D(P(S/sub n/)/spl par/Po(/spl lambda/))/spl les/1//spl lambda/ /spl Sigma//sub i=1//sup n/ ((p/sub i//sup 3/)/(1-p/sub i/)) and it is also generalized to the case when the X/sub i/ are general integer-valued random variables. Its proof is based on the derivation of a subadditivity property for a new discrete version of the Fisher information, and uses a recent logarithmic Sobolev inequality for the Poisson distribution.     

Random properties of the highest level sequences of primitive sequences over Z(2/sup e/)

Using the estimates of the exponential sums over Galois rings, we discuss the random properties of the highest level sequences /spl alpha//sub e-1/ of primitive sequences generated by a primitive polynomial of degree n over Z(2/sup e/). First we obtain an estimate of 0, 1 distribution in one period of /spl alpha//sub e-1/. On the other hand, we give an estimate of the absolute value of the autocorrelation function |C/sub N/(h)| of /spl alpha//sub e-1/, which is less than 2/sup e-1/(2/sup e-1/-1)/spl radic/3(2/sup 2e/-1)2/sup n/2/+2/sup e-1/ for h/spl ne/0. Both results show that the larger n is, the more random /spl alpha//sub e-1/ will be.     

The empirical distribution of rate-constrained source codes

Let X = (X/sub 1/,...) be a stationary ergodic finite-alphabet source, X/sup n/ denote its first n symbols, and Y/sup n/ be the codeword assigned to X/sup n/ by a lossy source code. The empirical kth-order joint distribution Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rceil/(x/sup k/,y/sup k/) is defined as the frequency of appearances of pairs of k-strings (x/sup k/,y/sup k/) along the pair (X/sup n/,Y/sup n/). Our main interest is in the sample behavior of this (random) distribution. Letting I(Q/sup k/) denote the mutual information I(X/sup k/;Y/sup k/) when (X/sup k/,Y/sup k/)/spl sim/Q/sup k/ we show that for any (sequence of) lossy source code(s) of rate /spl les/R lim sup/sub n/spl rarr//spl infin//(1/k)I(Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rfloor/) /spl les/R+(1/k)H (X/sub 1//sup k/)-H~(X) a.s. where H~(X) denotes the entropy rate of X. This is shown to imply, for a large class of sources including all independent and identically distributed (i.i.d.). sources and all sources satisfying the Shannon lower bound with equality, that for any sequence of codes which is good in the sense of asymptotically attaining a point on the rate distortion curve Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rfloor//spl rArr//sup d/P(X/sup k/,Y~/sup k/) a.s. whenever P(/sub X//sup k//sub ,Y//sup k/) is the unique distribution attaining the minimum in the definition of the kth-order rate distortion function. Consequences of these results include a new proof of Kieffer's sample converse to lossy source coding, as well as performance bounds for compression-based denoisers.     

On The General Relation Between /spl alpha/ and Q (Correspondence)

The relation between the quality factor Q and the attenuation constant /spl alpha/ of a transmission line has been known as follows: /spl alpha/ = /spl beta/ / /2Q where /spl beta/ is the phase constant. Recently from the following relation of propagation constant at resonance /spl Gamma/(/spl omega//sub 0/) + /spl part//spl Gamma/ / /spl part//spl omega/ /spl Delta//spl omega//spl cong/ i/spl beta/(/spl omega//sub 0/), where /spl Gamma/(/spl omega//sub 0/) = /spl alpha/(/spl omega//sub 0/) + i/spl beta/(/spl omega//sub 0/). Yeh derived a general relation between Q and /spl alpha/, namely, /spl alpha/ = /spl upsi//sub p/ / /spl upsi//sub g/ /spl beta/ / /2Q where /spl upsi//sub p/, and /spl upsi//sub g/ are the phase velocity and group velocity of the wave respectively. This general relation can be derived very simply from the generally accepted definition of /spl alpha/ and Q.     

Reduced-state representations for trellis codes using constellation symmetry

This paper presents a symmetry-based technique for trellis-code state-diagram reduction that has more general applicability than the quasi-regularity technique of Rouanne et al. and Zehavi et al. for trellis codes using standard constellations and labelings. For a 2/sup /spl nu/x/-state trellis code, the new technique reduces the 2/sup 2/spl nu/x/ state diagram to 2/sup /spl nu/x+/spl nu/q/-state diagram where 0/spl les//spl nu//sub q//spl les//spl nu//sub x/. The particular value of /spl nu//sub q/ depends on the constellation labeling and the convolutional encoder. For standard rate-k/(k+1) set-partitioned trellis codes, /spl nu//sub q/=0, and the overall number of states is the same with the new technique as with quasi-regularity. For codes that are not quasi-regular (and thus not amenable to the quasi-regularity technique), the new technique often provides some improvement (when /spl nu//sub q/相似文献