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Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are still under investigation. This paper covers some of the recent developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments. In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials (when codewords are viewed as evaluation vectors of degree r polynomials in m variables). It then overviews some of the algorithms for decoding RM codes. It covers both algorithms with provable performance guarantees for every block length, as well as algorithms with state-of-the-art performances in practical regimes, which do not perform as well for large block length. Finally, the paper concludes with a few open problems. [ABSTRACT FROM AUTHOR]

This paper shows that for any random variables $X$ and $Y$ , it is possible to represent $Y$ as a function of $(X,Z)$ such that $Z$ is independent of $X$ and $I(X;Z|Y)\le \log (I(X;Y)+1)+4$ bits. We use this strong functional representation lemma (SFRL) to establish a bound on the rate needed for one-shot exact channel simulation for general (discrete or continuous) random variables, strengthening the results by Harsha et al. and Braverman and Garg, and to establish new and simple achievability results for one-shot variable-length lossy source coding, multiple description coding, and Gray–Wyner system. We also show that the SFRL can be used to reduce the channel with state noncausally known at the encoder to a point-to-point channel, which provides a simple achievability proof of the Gelfand–Pinsker theorem. [ABSTRACT FROM AUTHOR]

This paper introduces Manhattan sampling in two and higher dimensions, and proves sampling theorems for them. In 2-D, Manhattan sampling, which takes samples densely along a Manhattan grid of lines, can be viewed as sampling on the union of two rectangular lattices, one dense horizontally and the other vertically, with the coarse spacing of each being a multiple of the fine spacing of the other. The sampling theorem shows that the images bandlimited to the union of the Nyquist regions of the two rectangular lattices can be recovered from their Manhattan samples, and an efficient procedure for doing so is given. Such recovery is possible even though there is an overlap among the spectral replicas induced by Manhattan sampling. In three and higher dimensions, there are many possible configurations for Manhattan sampling, each consisting of the union of special rectangular lattices called bi-step lattices. This paper identifies them, proves a sampling theorem showing that the images bandlimited to the union of the Nyquist regions of the bi-step rectangular lattices are recoverable from Manhattan samples, presents an efficient onion-peeling procedure for doing so, and shows that the union of Nyquist regions is as large as any bandlimited region, such that all images supported by such can be stably reconstructed from samples taken at the rate of the Manhattan sampling. It also develops a special representation for the bi-step lattices with a number of useful properties. While most of this paper deals with continuous-space images, Manhattan sampling of discrete-space images is also considered, for infinite, as well as finite, support images. [ABSTRACT FROM AUTHOR]

This paper considers the signal-to-noise ratio (SNR)-evolution, meaning the behavior as a function of the SNR, of the minimum mean-square error (MMSE) function of code sequences in several multi-user settings in the additive white Gaussian noise regime. The settings investigated in this context include the Gaussian wiretap channel, the Gaussian broadcast channel (BC), and the Gaussian BC with confidential messages (BCC). This paper shows that the specific properties of the SNR-evolution of the MMSE and conditional MMSE functions are necessary and sufficient conditions for capacity or equivocation achieving code sequences. In some cases, the complete SNR-evolution of a family of code sequences can be determined, providing significant insight into the disturbance (in terms of MMSE) such codes have on unintended receivers at other SNRs. Moreover, the effects of an additional MMSE constraint on the capacity region and on the SNR-evolution of code sequences are considered in the BC and BCC settings. Such an analysis emphasizes the tradeoff between rates and limited disturbance on unintended receivers. [ABSTRACT FROM AUTHOR]

Many problems in areas such as compressive sensing and coding theory seek to design a set of equal-norm vectors with large angular separation. This idea is essentially equivalent to constructing a frame with low coherence. The elements of such frames can in turn be used to build high-performance spherical codes, quantum measurement operators, and compressive sensing measurement matrices, to name a few applications. In this paper, we allude to the group-frame construction first described by Slepian and further explored in the works of Vale and Waldron. We present a method for selecting representations of a finite group to construct a group frame that achieves low coherence. Our technique produces a tight frame with a small number of distinct inner product values between the frame elements, in a sense approximating a Grassmannian frame. We identify special cases in which our construction yields some previously known frames with optimal coherence meeting the Welch lower bound, and other cases in which the entries of our frame vectors come from small alphabets. In particular, we apply our technique to the problem choosing a subset of rows of an Hadamard matrix so that the resulting columns form a low-coherence frame. Finally, we give an explicit calculation of the average coherence of our frames, and find regimes in which they satisfy the strong coherence property described by Mixon, Bajwa, and Calderbank. [ABSTRACT FROM AUTHOR]

Partial MDS (PMDS) codes are erasure codes combining local (row) correction with global additional correction of entries, while sector-disk (SD) codes are erasure codes that address the mixed failure mode of current redundant arrays of independent disk (RAID) systems. It has been an open problem to construct general codes that have the PMDS and the SD properties, and previous work has relied on Monte-Carlo searches. In this paper, we present a general construction that addresses the case of any number of failed disks and in addition, two erased sectors. The construction requires a modest field size. This result generalizes previous constructions extending RAID 5 and RAID 6. [ABSTRACT FROM AUTHOR]