on the residual entropy of the beg model at the antiquadrupolar ferromagnetic coexistence line we show that the residual entropy s for the two dimensional blume emery griffiths model at the antiquadrupolar ferromagnetic coexistence line satisfies the following bounds ln lambda 1 2n lambda 1 2n 1 leq s leq ln lambda 1 k mathit free k for all n 2 and k 1 where 1 n free and 1 n are the largest eigenvalues of the transfer matrices fn free and fn respectively in particular we have s 0 439396 0 008670

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

towards finding exact residual entropies of the ising antiferromagnets abstract we investigate various methods to evaluate exact residual entropies of the ising antiferromagnets in the maximum critical field on infinite lattices starting from results established for growing finite size lattices for the square ising antiferromagnet we obtain residual entropy with 14 correct digits 0 40749510126068

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

rank deficient solutions for optimal signaling over wiretap mimo channels capacity achieving signaling strategies for the gaussian wiretap multiple input multple output mimo channel are investigated without the degradedness assumption in addition to known solutions a number of new rank deficient solutions for the optimal transmit covariance matrix are obtained the case of a weak eavesdropper is considered in detail and the optimal covariance is established in an explicit closed form with no extra assumptions this provides lower and upper bounds to the secrecy capacity in the general case with a bounded gap which are tight for a weak eavesdropper or and low snr closed form solutions are also obtained for isotropic and omnidirectional eavesdroppers based on which lower and upper bounds to the secrecy capacity are established in the general case sufficient and necessary conditions for the optimality of three popular transmission techniques namely the zero forcing zf the standard water filling over the channel eigenmodes and the isotropic signaling is are established for the mimo wiretap channel these solutions are appealing due to their lower complexity in particular no wiretap codes are needed for the zf transmission and no precoding or feedback is needed for the isotropic signaling

on the ergodic secret key agreement over spatially correlated multiple antenna channels with public discussion we consider secret key agreement with public discussion over multiple input multiple output mimo rayleigh fast fading channels under correlated environment we assume that transmit legitimate receiver and eavesdropper antennas are correlated the legitimate receiver and the eavesdropper are assumed to have perfect channel knowledge while the transmitter has only knowledge of the correlation matrices first we derive the expression of the secret key capacity under the considered setup we prove that the optimal transmit strategy achieving the secret key capacity consists in transmitting independent gaussian signals along the eingenvectors of the transmit correlation matrix the powers allocated to each channel mode are determined as the solution to a numerical optimization problem a necessary and sufficient condition for beamforming i e transmitting along the strongest channel mode to be capacity achieving is derived moreover we analyze the impact of correlation matrices on the system performance finally we study the system s performance in the two extreme power regimes in the high power regime we provide closed form expressions of the gain loss due to correlation in the low signal to noise ratio snr regime we investigate the energy efficiency of the system by determining the minimum energy required for sharing a secret key bit and the wideband slope while highlighting the impact of correlation matrices

the worst additive noise under a covariance constraint the maximum entropy noise under a lag p autocorrelation constraint is known by burg s theorem to be the pth order gauss markov process satisfying these constraints the question is what is the worst additive noise for a communication channel given these constraints is it the maximum entropy noise the problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints r sub 0 r sub p for high signal powers the worst additive noise is gauss markov of order p as expected but for low powers the worst additive noise is gaussian with a covariance matrix in a convex set which depends on the signal power

uplink downlink duality via minimax duality the sum capacity of a gaussian vector broadcast channel is the saddle point of a minimax gaussian mutual information expression where the maximization is over the set of transmit covariance matrices subject to a power constraint and the minimization is over the set of noise covariance matrices subject to a diagonal constraint this sum capacity result has been proved using two different methods one based on decision feedback equalization and the other based on a duality between uplink and downlink channels this paper illustrates the connection between the two approaches by establishing that uplink downlink duality is equivalent to lagrangian duality in minimax optimization this minimax lagrangian duality relation allows the optimal transmit covariance and the least favorable noise covariance matrices in a gaussian vector broadcast channel to be characterized in terms of the dual variables in particular it reveals that the least favorable noise is not unique further the new lagrangian interpretation of uplink downlink duality allows the duality relation to be generalized to gaussian vector broadcast channels with arbitrary linear constraints however duality depends critically on the linearity of input constraints duality breaks down when the input constraint is an arbitrary convex constraint this shows that the minimax representation of the broadcast channel sum capacity is more general than the uplink downlink duality representation

the capacity region of the two receiver gaussian vector broadcast channel with private and common messages a novel method for establishing the optimality of gaussian auxiliary random variables in multiterminal information theory problems is developed this method is then employed to show that marton s inner bound achieves the capacity region of the two receiver gaussian vector broadcast channel with private and common messages

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

secure mimo communications under quantized channel feedback in the presence of jamming we consider the problem of secure communications in a mimo setting in the presence of an adversarial jammer equipped with nj transmit antennas and an eavesdropper equipped with ne receive antennas a multiantenna transmitter equipped with nt antennas desires to secretly communicate a message to a multiantenna receiver equipped with nr antennas we propose a transmission method based on artificial noise and linear precoding and a two stage receiver method employing beamforming under this strategy we first characterize the achievable secrecy rates of communication and prove that the achievable secure degrees of freedom sdof is given by ds nr nj in the perfect channel state information csi case second we consider quantized csi feedback using grassmannian quantization of a function of the direct channel matrix and derive sufficient conditions for the quantization bit rate scaling as a function of transmit power for maintaining the achievable sdof ds with perfect csi and for having asymptotically zero secrecy rate loss due to quantization numerical simulations are also provided to support the theory

secure transmission with multiple antennas part ii the mimome wiretap channel the capacity of the gaussian wiretap channel model is analyzed when there are multiple antennas at the sender intended receiver and eavesdropper the associated channel matrices are fixed and known to all the terminals a computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem the converse is based on a sato type argument used in other broadcast settings and the coding theorem is based on gaussian wiretap codebooks at high signal to noise ratio snr the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition and independently coding across the resulting parallel channels the associated capacity is expressed in terms of the corresponding generalized singular values it is shown that a semi blind masked multi input multi output mimo transmission strategy that sends information along directions in which there is gain to the intended receiver and synthetic noise along directions in which there is not can be arbitrarily far from capacity in this regime necessary and sufficient conditions for the secrecy capacity to be zero are provided which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed the resulting scaling laws establish that to prevent secure communication the eavesdropper needs three times as many antennas as the sender and intended receiver have jointly and that the optimum division of antennas between sender and intended receiver is in the ratio of 2 1

secure diversity multiplexing tradeoff of zero forcing transmit scheme at finite snr in this paper we address the finite signal to noise ratio snr diversity multiplexing tradeoff dmt of the multiple input multiple output mimo wiretap channel where a zero forcing zf transmit scheme that intends to send the secret information in the orthogonal space of the eavesdropper channel is used first we introduce the secrecy multiplexing gain at finite snr that generalizes the definition at high snr then we provide upper and lower bounds on the outage probability under secrecy constraint from which secrecy diversity gain estimates of zf are derived through asymptotic analysis we show that the upper bound underestimates the secrecy diversity gain whereas the lower bound is tight at high snr and thus its related diversity gain estimate is equal to the actual asymptotic secrecy diversity gain of the mimo wiretap channel

secure degrees of freedom of the gaussian diamond wiretap channel in this paper we consider the gaussian diamond wiretap channel that consists of an orthogonal broadcast channel from a source to two relays and a gaussian fast fading multiple access wiretap channel from the two relays to a legitimate destination and an eavesdropper for the multiple access part we consider both the case with full channel state information csi and the case with no eavesdropper s csi at the relays and the legitimate destination for both the cases we establish the exact secure degrees of freedom and generalize the results for multiple relays our results show 1 how to strike a balance between sending message symbols and common noise symbols from the source to the relays in the broadcast component and 2 how to combine artificial noise beamforming and noise alignment techniques at the relays in the multiple access component in the case with full csi we propose a scheme where the relays simultaneously beamform common noise signals in the null space of the legitimate destination s channel and align them with the message signals at the eavesdropper in the case with no eavesdropper s csi we present a scheme that efficiently utilizes the broadcast links by incorporating computation between the message and common noise symbols at the source finally most of our achievability and converse techniques can also be adapted to the gaussian non fading channel model

full rank solutions for the mimo gaussian wiretap channel with an average power constraint this paper considers a multiple input multiple output mimo gaussian wiretap channel with a transmitter a legitimate receiver and an eavesdropper each equipped with multiple antennas we first study the rank of the optimal input covariance matrix that achieves the secrecy capacity of the mimo gaussian wiretap channel under an average power constraint the rank and other properties of the optimal solution are derived based on certain relationships between the channel matrices for the legitimate receiver and eavesdropper next by obtaining necessary and sufficient conditions on the mimo wiretap channel parameters we determine the conditions under which the optimal input covariance matrix is full rank or rank deficient for the case that the optimal input covariance is full rank we fully characterize the solution when the optimal input covariance is rank deficient we show that the given mimo wiretap channel can be modeled by an equivalent wiretap channel whose optimal input covariance is full rank and achieves the same secrecy capacity as the original system numerical results are presented to illustrate the proposed theoretical findings

secret key agreement over non coherent block fading channels with public discussion motivated by recent interest in physical layer secret key generation over wireless fading channels we study the non coherent secret key generation capacity of a block fading wireless channel with channel reciprocity and bi directional two way communication we assume a non coherent main channel i e the realization of channel gains on the main channel is not known to any terminal the eavesdropper is assumed to have both perfect channel state information of its own channel and orthogonal observations from the forward and backward channels as our main result we establish new upper and lower bounds on the secret key generation capacity with public discussion which are structurally similar the upper bound can be expressed as a sum of three terms one of the terms arises due to channel reciprocity while the other two terms correspond to communication on the forward and backward channels respectively in the limit of long coherence period the contribution from channel reciprocity vanishes to zero whereas the other terms prevail the lower bound is based on a separation based scheme in each coherence block we use the first symbol for training while the remainder of the coherence block is devoted to source emulation i e to generate correlated sources between the terminals the lower bound expression also consists of three terms and admits an interpretation similar to the upper bound expression for rayleigh fading channels in the high signal to noise ratio snr regime the gap between the upper and lower bounds is shown to vanish inversely with the coherence period numerical results indicate significant performance gains over training only schemes even for moderate values of snr and small coherence periods

gradient of mutual information in linear vector gaussian channels this paper considers a general linear vector gaussian channel with arbitrary signaling and pursues two closely related goals i closed form expressions for the gradient of the mutual information with respect to arbitrary parameters of the system and ii fundamental connections between information theory and estimation theory generalizing the fundamental relationship recently unveiled by guo shamai and verdu spl acute we show that the gradient of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the best estimate of the input given the output gradients and derivatives with respect to other parameters are then found via the differentiation chain rule

algorithms for secrecy guarantee with null space beamforming in two way relay networks in this paper we consider a two way relay network with two sources and multiple cooperative relays in the presence of an eavesdropper to guarantee the secrecy a null space beamforming scheme is applied where the relay beamforming vector lies in the nullspace of the equivalent channel of relay link from two sources to the eavesdropper our goal is to obtain the optimal beamforming vector as well as two sources transmit power subject to various criteria we propose three different approaches and solve them in an alternating iterative way where subproblems for solving beamforming and sources power are formulated in each iteration respectively first we minimize the total transmit power under secrecy rate constraint at two sources for beamforming vector subproblem two different methods semi definite programming sdp and sequential quadratic programming sqp are proposed where we analyze and verify sqp has lower complexity than sdp second we maximize the secrecy sum rate subject to total transmit power constraint the beamforming vector subproblem is equivalent to a generalized rayleigh quotient problem with rank constraint third the problem of minimum per user secrecy rate maximization under the total power constraint is investigated for user fairness an iterative procedure utilizing the sdp with bisection search method is proposed to solve beamforming subproblem in each approach the subproblem with two sources power is formulated as a single variable problem and solved by newton s method with line search simulation results demonstrate the validity of proposed approaches and algorithms for both symmetric and asymmetric scenarios

a percolation system with extremely long range connections and node dilution abstract we study the very long range bond percolation problem on a linear chain with both node and bond dilution very long range means that the probability p i j for a connection between two nodes i j at a distance r i j decays as a power law i e p i j r i j n 1 when 0 1 and p i j r i j ln n when 1 node dilution means that the probability that a node is present in a site is p s 0 1 the behavior of this model results from the competition between long range connectivity which enhances the percolation and node dilution which weakens percolation the case 0 with p s 1 is well known being the exactly solvable mean field model the percolation order parameter p is investigated numerically for different values of p s and we show that in all range 0 1 the percolation order parameter p depends only on the average connectivity of the nodes which can be explicitly computed in terms of the three parameters p s and

on the residual entropy of the blume emery griffiths model by means of a transfer matrix method we show that the residual entropy s of the two dimensional square lattice blume emery griffiths model on the antiquadrupolar disordered and ferromagnetic antiquadrupolar disordered phase boundaries satisfies the inequalities ln 1 n n 1 s ln 1 n n where 1 n is the largest eigenvalue of a transfer matrix fn on a strip of width n these bounds imply the existence of a o 1 n correction in the approximation of s by ln 1 n n using these bounds we calculate numerically the value of s with precise estimates on the errors

a remark on the residual entropy of the antiferromagnetic ising model in the maximal critical field by means of a transfer matrix method we show that the residual entropy s of the two dimensional square lattice antiferromagnetic ising model in the maximal critical field satisfies ln n n 1 s ln n n where n is the largest eigenvalue of the transfer matrix fn on a strip of width n using these bounds we numerically calculate the value of s with precise estimates on the errors namely s 0 394198 0 020747

on secrecy rate analysis of mimo wiretap channels driven by finite alphabet input this work investigates the effect of finite alphabet input constraint on the secrecy rate of a multi antenna wiretap channel most existing works have characterized maximum achievable secrecy rate or secrecy capacity for single and multiple antenna systems based on gaussian source signals and secrecy code for practical considerations we study the effect of finite discrete constellation on the achievable secrecy rate of multiple antenna wire tap channels our proposed precoding scheme converts the underlying multi antenna system into a bank of parallel channels based on this precoding strategy we develop a decentralized power allocation algorithm based on dual decomposition to maximize the achievable secrecy rate in addition we analyze the achievable secrecy rate for finite alphabet inputs in low and high snr regions our results demonstrate substantial difference in secrecy rate between systems given finite alphabet inputs and systems with gaussian inputs

optimal signaling for secure communications over gaussian mimo wiretap channels optimal signaling over the gaussian multiple input multiple output wire tap channel is studied under the total transmit power constraint a closed form solution for an optimal transmit covariance matrix is obtained when the channel is strictly degraded in combination with the rank 1 solution this provides the complete characterization of the optimal covariance for the case of two transmit antennas the cases of weak eavesdropper and high snr are considered it is shown that the optimal covariance does not converge to a scaled identity in the high snr regime necessary optimality conditions and a tight upper bound on the rank of an optimal covariance matrix are established for the general case along with a lower bound to the secrecy capacity which is tight in a number of scenarios

secure communications for dual polarized mimo systems to enhance secure communications we deploy the dual polarized antenna arrays at communication nodes of the multi input multioutput mimo system where the base station communicates with multiple legitimate users in the presence of an eavesdropper we also adopt the dual structured precoding in which a preprocessing matrix based on the polarized array spatial correlation and a linear precoding based on the instantaneous channel state information csi are concatenated we design this dual structured multiuser linear precoding under three cases in the first case given perfect global csi the secrecy rate optimization problem is formulated and transformed into the weighted minimum mean square error mse problem which can be effectively solved by the block coordinate decent method in the second case where the eavesdropper s csi is unavailable an artificial noise is generated to confuse the eavesdropper by minimizing the information transmit power subject to a preset mse threshold for the recovered confidential signals which can be solved by an efficient iterative algorithm in the third case of imperfect global csi the robust optimization for secure communications is performed by minimizing the largest received mse among the users subject to the total transmit power constraint which can be reformulated into a biconvex semidefinite programming problem and solved by an efficient alternating convex optimization simulation results are included to demonstrate the excellent performance of our proposed designs over the conventional single polarized array based designs in terms of achievable secrecy rate minimum transmit power and the mse of recovered confidential signals

upper critical dimension of the negative weight percolation problem by means of numerical simulations we investigate the geometric properties of loops on hypercubic lattice graphs in dimensions d 2 through 7 where edge weights are drawn from a distribution that allows for positive and negative weights we are interested in the appearance of system spanning loops of total negative weight the resulting negative weight percolation nwp problem is fundamentally different from conventional percolation as we have seen in previous studies of this model for the two dimensional case here we characterize the transition for hypercubic systems where the aim of the present study is to get a grip on the upper critical dimension d u of the nwp problem for the numerical simulations we employ a mapping of the nwp model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms we characterize the loops via observables similar to those in percolation theory and perform finite size scaling analyses e g three dimensional hypercubic systems with side length up to l 56 sites in order to estimate the critical properties of the nwp phenomenon we find our numerical results consistent with an upper critical dimension d u 6 for the nwp problem

largest and second largest cluster statistics at the percolation threshold of hypercubic lattices we investigate the scale invariance of the average ratio between the masses of the largest and second largest clusters at percolation we employ a finite size scaling method to estimate percolation thresholds based on the simulations of relatively small lattices and report on estimates for p c in hypercubic lattices with d 2 7 in full agreement with the best literature estimates also we find the critical mass ratio to be strongly dependent on the boundary conditions decreasing with the lattice dimension further we compute several relevant mass distribution functions associated with the two largest clusters which approach to limiting distributions for dg6 finally we discuss the main relevant features of the mass distributions in light of the relative role played by the spanning and nonspanning clusters

the secrecy capacity of the mimo wiretap channel we consider the mimo wiretap channel that is a mimo broadcast channel where the transmitter sends some confidential information to one user which is a legitimate receiver while the other user is an eavesdropper perfect secrecy is achieved when the transmitter and the legitimate receiver can communicate at some positive rate while insuring that the eavesdropper gets zero bits of information in this paper we compute the perfect secrecy capacity of the multiple antenna mimo broadcast channel where the number of antennas is arbitrary for both the transmitter and the two receivers our technique involves a careful study of a sato like upper bound via the solution of a certain algebraic riccati equation

an algorithm for global maximization of secrecy rates in gaussian mimo wiretap channels optimal signaling for secrecy rate maximization in gaussian mimo wiretap channels is considered while this channel has attracted a significant attention recently and a number of results have been obtained including the proof of the optimality of gaussian signalling an optimal transmit covariance matrix is known for some special cases only and the general case remains an open problem an iterative custom made algorithm to find a globally optimal transmit covariance matrix in the general case is developed in this paper with guaranteed convergence to a global optimum while the original optimization problem is not convex and hence difficult to solve its minimax reformulation can be solved via the convex optimization tools which is exploited here the proposed algorithm is based on the barrier method extended to deal with a minimax problem at hand its convergence to a global optimum is proved for the general case degraded or not and a bound for the optimality gap is given for each step of the barrier method the performance of the algorithm is demonstrated via numerical examples in particular 20 to 40 newton steps are already sufficient to solve the sufficient optimality conditions with very high precision up to the machine precision level even for large systems even fewer steps are required if the secrecy capacity is the only quantity of interest the algorithm can be significantly simplified for the degraded channel case and can also be adopted to include the per antenna power constraints instead or in addition to the total power constraint it also solves the dual problem of minimizing the total power subject to the secrecy rate constraint

energy efficient secure communication over decode and forward relay channels in this paper we raise the concern on energy efficient secure communication over a decode and forward relay network with potential eavesdroppers our objective is maximizing the secure energy efficiency ee which is defined as the ratio of secrecy rate to total power subject to the given individual power relay decoding rate and target secrecy rate constraints the problem is formulated as a unified mixed integer nonlinear optimization adapting to different assumptions of channel state information csi to deal with this combinatorial optimization a suboptimal solution scheme is developed upon some mathematical methods as decoupling of mixed integer programming fractional programming dual decomposition and difference of convex functions programming the core of the scheme is to transform the primal problem into simple subproblems step by step and then convex programming can be exploited finally the proposed scheme for the case with statistical eavesdropper s csi may be conditionally extended to the scenario with full csi in which a local optimal solution may be obtained in certain cases finally the performance and achievable secure ee of the proposed algorithm are demonstrated by simulations where the tradeoff between secure ee and secrecy rate is also revealed

on the secrecy capacity of the wiretap channel with imperfect main channel estimation we study the secrecy capacity of fast fading channels under imperfect main channel between the transmitter and the legitimate receiver estimation at the transmitter lower and upper bounds on the ergodic secrecy capacity are derived for a class of independent identically distributed i i d fading channels the achievable rate follows from a standard wiretap code in which a simple on off power control is employed along with a gaussian input the upper bound is obtained using an appropriate correlation scheme of the main and eavesdropper channels and is the best known upper bound so far the upper and lower bounds coincide with recently derived ones in case of perfect main csi furthermore the upper bound is tight in case of no main csi where the secrecy capacity is equal to zero asymptotic analysis at high and low signal to noise ratio snr is also given at high snr we show that the capacity is bounded by providing upper and lower bounds that depend on the channel estimation error at low snr however we prove that the secrecy capacity is asymptotically equal to the capacity of the main channel as if there were no secrecy constraint numerical results are provided for i i d rayleigh fading channels

mimo broadcast channel with an unknown eavesdropper secrecy degrees of freedom we study a multi antenna broadcast channel with two legitimate receivers and an external eavesdropper we assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint as our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages we show that a direct extension of the single user wiretap codebook does not achieve the secrecy degrees of freedom our proposed optimal scheme involves decomposing the signal space into a common subspace which can be observed by both receivers and private subspaces which can be observed by only one of the receivers and carefully transmitting a subset of messages in each subspace we also consider the case when each user s private message must additionally remain confidential from the other legitimate receiver and characterize the s d o f region in this case

secure communication in the low snr regime secrecy capacity of a multiple antenna wiretap channel is studied in the low signal to noise ratio snr regime expressions for the first and second derivatives of the secrecy capacity with respect to snr at snr 0 are derived transmission strategies required to achieve these derivatives are identified in particular it is shown that it is optimal in the low snr regime to transmit in the maximal eigenvalue eigenspace of hm hm nm ne he he where hm and he denote the channel matrices associated with the legitimate receiver and eavesdropper respectively and nm and ne are the noise variances at the receiver and eavesdropper respectively energy efficiency is analyzed by finding the minimum bit energy required for secure and reliable communications and the wideband slope increased bit energy requirements under secrecy constraints are quantified finally the impact of fading is investigated and the benefits of fading in terms of energy efficiency are shown

residual entropy of the square ising antiferromagnet in the maximum critical field the fibonacci matrix we find the analytical expression for the residual entropy of the square ising model with nearest neighbour antiferromagnetic coupling j in the maximum critical field in terms of the fibonacci matrix which itself represents a self similar fractal object the result coincides with the existing numerical data by considering regular self similar fractal objects rather than seemingly random transfer matrices this approach opens the possibility of finding the corresponding solutions in more complicated cases such as the antiferromagnets with longer than nearest neighbour interactions and the three dimensional antiferromagnets as well as the possibility of unification of results pertinent to different lattices in two and three dimensions