Universally optimal (UO) codes were introduced by H. Cohn and A. Kumar in 2007 and later extended to the discrete setting by Cohn and Y. Zhao. They minimize the ``energy'' among all codes of the same size for a certain class of potential functions. So far only a small number of specific functionals have been linked to information theory problems. We add one more, showing that UO codes optimize $α$-mutual information for $α=2$ in the context of wiretap channels with a noiseless main channel. This implies that UO codes are optimal for transmission over this type of wiretap channels.

We improve the secrecy guarantees for transmission over BMS wiretap channels that relies on regular LDPC codes. Previous works (Thangaraj e.a., 2007, 2010) showed that LDPC codes achieve secrecy capacity of some classes of wiretap channels while leaking$o(n)$bits of information over$n$uses of the channel. We improve the security component of these results by reducing the leakage parameter to$O\left(\log ^{2} n\right)$. While this result stops short of proving strong security, it goes beyond the general uniformity properties of capacity-approaching code families.

We consider the problem of distilling uniform random bits from an unknown source with a givenp-entropy using linear hashing. As our main result, we estimate the expectedp-divergence from the uniform distribution over the ensemble of random linear codes for all integerp≥ 2. The proof relies on analyzing how additive noise, determined by a random element of the code from the ensemble, acts on the source distribution. This action leads to the transformation of the source distribution into an approximately uniform one, a process commonly referred to as distribution smoothing. We also show that hashing with Reed-Muller matrices reaches intrinsic randomness of memoryless Bernoulli sources in thelpsense for all integerp≥ 2.

Universal hash functions map the output of a source to random strings over a finite alphabet, aiming to approximate the uniform distribution on the set of strings. A classic result on these functions, called the Leftover Hash Lemma, gives an estimate of the distance from uniformity based on the assumptions about the min-entropy of the source. We prove several results concerning extensions of this lemma to a class of functions that arek*-universal, i.e.,l-universal for all 2 ≤l≤k. As a common distinctive feature, our results provide estimates of closeness to uniformity in terms of the α-Rényi divergence for all α ∈ (1, ∞]. For 1 ≤ α ≤kwe show that it is possible to convert all the randomness of the source measured in α-Rényi entropy into approximately uniform bits with nearly the same amount of randomness. For large enoughkwe show that it is possible to distill random bits that are nearly uniform, as measured by min-entropy. We also extend these results to hashing with side information.

Shared information is a measure of mutual dependence among$m\geq 2$jointly distributed discrete random variables. We show that the shared information of a Markov random field in which the underlying graph has at least one cut vertex, is the same as the minimum shared information of its blocks (also called biconnected components). This generalizes prior results on shared information of Markov random fields to a much wider class of nontree graphs.

There are several systems that use one or several Kinect sensors for human gait analysis, particularly for diagnosis of patients. However, due to the limited depth sensing range of the Kinect—a sensor manufactured for video gaming—the depth measurement accuracy reduces with distance from the Kinect. In addition, self-occlusion of the subject limits the accuracy and utility of such systems. We overcome these limitations by using a two-Kinect gait analysis system and mechanically moving the Kinects in synchronization with the test subject and each other. This increases the practical measurement range of the Kinect-based system whilst maintaining the measurement accuracy. Results of the comparison of knee flexion, step length, and stride length with a software based method show that our moving Kinect system can accurately analyse these gait parameters.