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This paper considers “ $\delta $ -almost Reed–Muller codes”, i.e., linear codes spanned by evaluations of all but a $\delta $ fraction of monomials of degree at most $d$. It is shown that for any $\delta > 0$ and any $\varepsilon >0$ , there exists a family of $\delta $ -almost Reed–Muller codes of constant rate that correct $1/2- \varepsilon $ fraction of random errors with high probability. For exact Reed–Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed–Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC ’15). Our proof is based on the recent polarization result for Reed–Muller codes, combined with a combinatorial approach to establishing inequalities between the Reed–Muller code entropies. [ABSTRACT FROM AUTHOR]

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]

Reed-Muller (RM) codes were introduced in 1954 and have long been conjectured to achieve Shannon’s capacity on symmetric channels. The activity on this conjecture has recently been revived with the emergence of polar codes. RM codes and polar codes are generated by the same matrix $\begin{aligned} G_{m}= \left[{\begin{smallmatrix}1 & 0 \\ 1 & 1 \\ \end{smallmatrix}}\right]^{\otimes m} \end{aligned}$ but using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in $G_{m}$ ; while this is a more elaborate and channel-dependent rule, the top-to-down ordering allows Arıkan to show that the conditional mutual information polarizes, and this gives directly a capacity-achieving code on any symmetric channel. RM codes are yet to be proved to have such a property, despite the recent success for the erasure channel. In this article, we connect RM codes to polarization theory. We show that proceeding in the RM code ordering, i.e., not top-to-down but from the lightest to the heaviest rows in $G_{m}$ , the conditional mutual information again polarizes. Here “polarization” means that almost all the conditional mutual information becomes either very close to 0 or very close to 1. Polarization itself is a necessary condition for RM codes to achieve capacity on symmetric channels while polarization together with a strong order on the conditional mutual information gives a sufficient condition, where strong order means that rows with larger weight always correspond to larger conditional mutual information. Although we are not able to prove the strong order, we establish a partial order on the conditional mutual information, which is a subset of the strong order. While the main results of this article–polarization together with the partial order–provide some advances on the capacity-achieving conjecture of RM codes, we emphasize that our results do not allow us to prove the conjecture. [ABSTRACT FROM AUTHOR]

We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are lower-order RM codes), and aggregating the reconstructions (e.g., using majority votes). We further provide extensions of the algorithms using list-decoding. We run our algorithm for AWGN channels and Binary Symmetric Channels at the short code length (≤ 1024) regime for a wide range of code rates. Simulation results show that in both low code rate and high code rate regimes, the new algorithm outperforms the widely used decoder for polar codes (SCL+CRC) with the same parameters. The performance of the new algorithm for RM codes in those regimes is in fact close to that of the maximal likelihood decoder. Finally, the new decoder naturally allows for parallel implementations. [ABSTRACT FROM AUTHOR]

The polarization phenomenon for a single source is extended to a framework with multiple correlated sources. It is shown in addition to extracting the randomness of the source, the polar transforms take the original arbitrary dependencies to extremal dependencies. Polar coding schemes for the Slepian–Wolf (SW) coding problem and for secret key generations are then proposed based on this phenomenon. In particular, secret keys achieving the secrecy capacity and compression schemes achieving the SW capacity region are obtained with a complexity of $O(n \log (n))$ . [ABSTRACT FROM AUTHOR]

Polar codes are introduced for discrete memoryless broadcast channels. For m -user deterministic broadcast channels, polarization is applied to map uniformly random message bits from m -independent messages to one codeword while satisfying broadcast constraints. The polarization-based codes achieve rates on the boundary of the private-message capacity region. For two-user noisy broadcast channels, polar implementations are presented for two information-theoretic schemes: 1) Cover’s superposition codes and 2) Marton’s codes. Due to the structure of polarization, constraints on the auxiliary and channel-input distributions are identified to ensure proper alignment of polarization indices in the multiuser setting. The codes achieve rates on the capacity boundary of a few classes of broadcast channels (e.g., binary-input stochastically degraded). The complexity of encoding and decoding is O(n \log n) , where n of the average block error probability where 0 < \beta < {1}/{2} . Reproducible experiments for finite block lengths n = 512, 1024, 2048$ corroborate the theory. [ABSTRACT FROM AUTHOR]

In this paper, polar codes for the m-user multiple access channel (MAC) with binary inputs are constructed. It is shown that Arikan's polarization technique applied individually to each user transforms independent uses of an m-user binary input MAC into successive uses of extremal MACs. This transformation has a number of desirable properties: 1) the “uniform sum-rate” of the original MAC is preserved, 2) the extremal MACs have uniform rate regions that are not only polymatroids but matroids, and thus, 3) their uniform sum-rate can be reached by each user transmitting either uncoded or fixed bits; in this sense, they are easy to communicate over. A polar code can then be constructed with an encoding and decoding complexity of O(n \log n) (where n is the block length), a block error probability of o(\exp (- n^1/2 - \varepsilon)), and capable of achieving the uniform sum-rate of any binary input MAC with arbitrary many users. Applications of this polar code construction to channels with a finite field input alphabet and to the additive white Gaussian noise channel are also discussed. [ABSTRACT FROM PUBLISHER]