Consider a binary-input memoryless outputsymmetric channel W . Such a channel has a capacity, call it I(W ), and for any R < I(W ) and strictly positive constant Pe we know that we can construct a coding scheme that allows transmission at rate R with an error probability not exceeding Pe. Assume now that we let the rate R tend to I(W ) and we ask how we have to "scale" the blocklength N in order to keep the error probability fixed to Pe. We refer to this as the "finitelength scaling" behavior. This question was addressed by Strassen as well as Polyanskiy, Poor and Verdu, and the result is that N must grow at least as the square of the reciprocal of I(W ) − R.Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Since the exact scaling behavior depends on the choice of the channel our objective is to provide scaling laws that hold universally for all binaryinput memoryless output-symmetric channels. Our approach is based on analyzing the dynamics of the un-polarized channels. More precisely, we provide bounds on (the exponent of) the number of sub-channels whose Bhattacharyya constant falls in a fixed interval [a, b]. Mathematically, this can be stated as bounding the sequence 1 n log Pr (Zn ∈ [a, b]) n∈N , where Zn is the Bhattacharyya process. We then use these bounds to derive trade-offs between the rate and the block-length.The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of sub-channels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value Pe > 0, then the required block-length N scales in terms of the rate R < I(W ) as N ≥ α (I(W )−R) µ , where α is a positive constant that depends on Pe and I(W ). We show that µ = 3.579 is a valid choice, and we conjecture that indeed the value of µ can be improved to µ = 3.627, the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like N ≤ β (I(W )−R) µ , where β is a constant that depends on Pe and I(W ), and µ = 6.

In this paper we introduce a new class of codes for over-loaded synchronous wireless and optical CDMA systems which increases the number of users for fixed number of chips without introducing any errors. Equivalently, the chip rate can be reduced for a given number of users, which implies bandwidth reduction for downlink wireless systems. An upper bound for the maximum number of users for a given number of chips is derived. Also, lower and upper bounds for the sum channel capacity of a binary over-loaded CDMA are derived that can predict the existence of such over-loaded codes. We also propose a simplified maximum likelihood method for decoding these types of over-loaded codes. Although a high percentage of the over-loading factor 3 degrades the system performance in noisy channels, simulation results show that this degradation is not significant. More importantly, for moderate values of ‫ܧ‬ ܰ Τ (in the range of -ͳͲ dB) or higher, the proposed codes perform much better than the binary Welch bound equality sequences.

The spherical ensemble is a well-studied determinantal process with a fixed number of points on S 2 . The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere.In this paper we study the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap. Moreover, we consider this process as a way of distributing points uniformly on the sphere. To this aim, we study two "metrics" to measure the uniformity of an arrangement of points on the sphere. For each of these metrics (discrepancy and Riesz energies) we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).for all continuous, compactly supported functions F : C k → C, where dµ(z) := n π(1+|z| 2 ) n+1 dz and dz denotes the Lebesgue measure on the complex plane C.Krishnapur [22] showed that this random point process is a determinantal point process on complex plane with kernel K (n) (z, w) = (1 + zw) n−1 with respect to the background measure dµ(z), i.e. we havefor every k ≥ 1 and z 1 , . . . , z k ∈ C. We note here that a random point process is said to be a determinantal point process if its k-point correlation functions have determinantal form similar to (1.2). The corresponding kernel K (n) (z, w) is called a correlation kernel of the determinantal point process. We refer to [20] or [21] and references therein for more information on deteminantal point processes. Let S 2 = {p ∈ R 3 : |p| = 1} be the unit two-dimensional sphere centred at the origin in three-dimensional Euclidean space R 3 . Also we let ν denotes the Lebesgue surface area measure on this sphere with total measure 4π. As mentioned in [21], these eigenvalues are best thought of as points on S 2 , using stereographic projection. Let g be the stereographic projection of the sphere S 2 from the north pole onto the plane {(t 1 , t 2 , 0); t 1 , t 2 ∈ R}. If we let P i = g −1 (λ i ) for 1 ≤ i ≤ n then the vector (P 1 , . . . , P n ), in uniform random order, has the joint density Const.i

This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, . . . . First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ −1 (1−e −λu ). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

In this paper, we obtain a family of lower bounds for the sum capacity of Code Division Multiple Access (CDMA) channels assuming binary inputs and binary signature codes in the presence of additive noise with an arbitrary distribution. The envelope of this family gives a relatively tight lower bound in terms of the number of users, spreading gain and the noise distribution. The derivation methods for the noiseless and the noisy channels are different but when the noise variance goes to zero, the noisy channel bound approaches the noiseless case. The behavior of the lower bound shows that for small noise power, the number of users can be much more than the spreading gain without any significant loss of information (overloaded CDMA). A conjectured upper bound is also derived under the usual assumption that the users send out equally likely binary bits in the presence of additive noise with an arbitrary distribution. As the noise level increases, and/or, the ratio of the number of users and the spreading gain increases, the conjectured upper bound approaches the lower bound. We have also derived asymptotic limits of our bounds that can be compared to a formula that Tanaka obtained using techniques from statistical physics; his bound is close to that of our conjectured upper bound for large scale systems.