An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets

We prove a lower estimate on the increase in entropy when two copies of a conditional random variable X | Y, with X supported on Z_q={0,1,...,q-1} for prime q, are summed modulo q. Specifically, given two i.i.d. copies (X_1,Y_1) and (X_2,Y_2) of a pair of random variables (X,Y), with X taking values in Z_q, we show H(X_1 + X_2 mid Y_1, Y_2) - H(X|Y) >=e alpha(q) * H(X|Y) (1-H(X|Y)) for some alpha(q) > 0, where H(.) is the normalized (by factor log_2(q)) entropy. In particular, if X | Y is not close to being fully random or fully deterministic and H(X| Y) in (gamma,1-gamma), then the entropy of the sum increases by Omega_q(gamma). Our motivation is an effective analysis of the finite-length behavior of polar codes, for which the linear dependence on gamma is quantitatively important. The assumption of q being prime is necessary: for X supported uniformly on a proper subgroup of Z_q we have H(X+X)=H(X). For X supported on infinite groups without a finite subgroup (the torsion-free case) and no conditioning, a sumset inequality for the absolute increase in (unnormalized) entropy was shown by Tao in [Tao, CP&R 2010]. We use our sumset inequality to analyze Ari kan\u27s construction of polar codes and prove that for any q-ary source X, where q is any fixed prime, and anyepsilon > 0, polar codes allow efficient data compression of N i.i.d. copies of X into (H(X)+epsilon)N q-ary symbols, as soon as N is polynomially large in 1/epsilon. We can get capacity-achieving source codes with similar guarantees for composite alphabets, by factoring q into primes and combining different polar codes for each prime in factorization. A consequence of our result for noisy channel coding is that for all discrete memoryless channels, there are explicit codes enabling reliable communication within epsilon > 0 of the symmetric Shannon capacity for a block length and decoding complexity bounded by a polynomial in 1/epsilon. The result was previously shown for the special case of binary-input channels [Guruswami/Xial, FOCS\u2713; Hassani/Alishahi/Urbanke, CoRR 2013], and this work extends the result to channels over any alphabet

On the Finite Length Scaling of Ternary Polar Codes

The polarization process of polar codes over a ternary alphabet is studied. Recently it has been shown that the scaling of the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this work, it is shown that a much lower degree polynomial can be computed numerically for the ternary case. Similar results are conjectured for the general case of prime alphabet size.Comment: Submitted to ISIT 201

R\'enyi Bounds on Information Combining

Bounds on information combining are entropic inequalities that determine how the information, or entropy, of a set of random variables can change when they are combined in certain prescribed ways. Such bounds play an important role in information theory, particularly in coding and Shannon theory. The arguably most elementary kind of information combining is the addition of two binary random variables, i.e. a CNOT gate, and the resulting quantities are fundamental when investigating belief propagation and polar coding. In this work we will generalize the concept to R\'enyi entropies. We give optimal bounds on the conditional R\'enyi entropy after combination, based on a certain convexity or concavity property and discuss when this property indeed holds. Since there is no generally agreed upon definition of the conditional R\'enyi entropy, we consider four different versions from the literature. Finally, we discuss the application of these bounds to the polarization of R\'enyi entropies under polar codes.Comment: 14 pages, accepted for presentation at ISIT 202

Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead to concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019 IEEE International Symposium on Information Theory (ISIT2019

Bounds on Information Combining With Quantum Side Information

"Bounds on information combining" are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in Belief propagation and Polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities. Our main technical result is a non-trivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal "quantum Mrs. Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the concavity of von Neumann entropy, which is tight in the regime of low pairwise state fidelities; (2) the quantitative improvement of strong subadditivity due to Fawzi-Renner, in which we manage to handle the minimization over recovery maps; (3) recent duality results on classical-quantum-channels due to Renes et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence. We finally apply our bounds to Polar coding for binary-input classical-quantum channels, and show the following three results: (A) Even non-stationary channels polarize under the polar transform. (B) The blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity. (C) Under the aforementioned lower bound conjecture, a blocklength polynomial in the gap suffices.Comment: 23 pages, 6 figures; v2: small correction

Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound

We introduce synchronization strings as a novel way of efficiently dealing with synchronization errors, i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to deal with than commonly considered half-errors, i.e., symbol corruptions and erasures. For every $\epsilon >0$, synchronization strings allow to index a sequence with an $\epsilon^{-O(1)}$ size alphabet such that one can efficiently transform $k$ synchronization errors into $(1+\epsilon)k$ half-errors. This powerful new technique has many applications. In this paper, we focus on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion deletion channels. While ECCs for both half-errors and synchronization errors have been intensely studied, the later has largely resisted progress. Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed by Schulman and Zuckerman. Insdel codes for asymptotically large or small noise rates were given in 2016 by Guruswami et al. but these codes are still polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A direct application of our synchronization strings based indexing method gives a simple black-box construction which transforms any ECC into an equally efficient insdel code with a slightly larger alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs over large constant alphabets into the realm of insdel codes. Most notably, we obtain efficient insdel codes which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound for the complete noise spectrum