Ryuhei Mori

Performance of polar codes with the construction using density evolution

Polar coding, proposed by Arikan, makes it possible to construct capacity-achieving codes for symmetric binary-input discrete memoryless channels, with low encoding and decoding complexity. Complexity of the originally proposed code construction method, however, grows exponentially in the block-length unless a channel is the binary erasure channel. Recently, the authors have proposed a new capacity-achieving code construction method with linear complexity in the block-length for arbitrary symmetric binary-input memoryless channels. In this letter, we evaluate performance of polar codes designed with the new construction method, and compare it with that of the codes constructed with another heuristic method with linear complexity proposed by Arikan.

Performance and construction of polar codes on symmetric binary-input memoryless channels

Channel polarization is a method of constructing capacity achieving codes for symmetric binary-input discrete memoryless channels (B-DMCs) [1]. In the original paper, the construction complexity is exponential in the blocklength. In this paper, a new construction method for arbitrary symmetric binary memoryless channel (B-MC) with linear complexity in the blocklength is proposed. Furthermore, new upper bound and lower bound of the block error probability of polar codes are derived for the BEC and arbitrary symmetric B-MC, respectively.

Channel polarization on q-ary discrete memoryless channels by arbitrary kernels

A method of channel polarization, proposed by Arikan, allows us to construct efficient capacity-achieving channel codes. In the original work, binary input discrete memoryless channels are considered. A special case of q-ary channel polarization is considered by Şaşoğlu, Telatar, and Arikan. In this paper, we consider more general channel polarization on q-ary channels. We further show explicit constructions using Reed-Solomon codes, on which asymptotically fast channel polarization is induced.

Non-binary polar codes using Reed-Solomon codes and algebraic geometry codes

Polar codes, introduced by Arıkan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes using Hermitian codes have asymptotically good performance.

Rate-Dependent Analysis of the Asymptotic Behavior of Channel Polarization

We consider the asymptotic behavior of the polarization process in the large block-length regime when transmission takes place over a binary-input memoryless symmetric channel W. In particular, we study the asymptotics of the cumulative distribution P(Zn ≤ z), where {Zn} is the Bhattacharyya process associated with W, and its dependence on the rate of transmission. On the basis of this result, we characterize the asymptotic behavior, as well as its dependence on the rate, of the block error probability of polar codes using the successive cancellation decoder. This refines the original asymptotic bounds by Arıkan and Telatar. Our results apply to general polar codes based on l×l kernel matrices. We also provide asymptotic lower bounds on the block error probability of polar codes using the maximum a posteriori (MAP) decoder. The MAP lower bound and the successive cancellation upper bound coincide when l = 2, but there is a gap for l > 2.

Source and Channel Polarization Over Finite Fields and Reed–Solomon Matrices

Polarization phenomenon over any finite field Fq with size q being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arıkan for the binary field, as well as its extension to a prime field by Sasoglu, Telatar, and Arıkan. In this paper, a necessary and sufficient condition of a matrix over a finite field Fq is shown under which any source and channel are polarized. Furthermore, the result of the speed of polarization for the binary alphabet obtained by Arıkan and Telatar is generalized to arbitrary finite field. It is also shown that the asymptotic error probability of polar codes is improved by using the Reed-Solomon matrices, which can be regarded as a natural generalization of the 2 × 2 binary matrix used in the original proposal by Arıkan.

Refined rate of channel polarization

A rate-dependent upper bound of the best achievable block error probability of polar codes with successive-cancellation decoding is derived.

Connection between annealed free energy and belief propagation on random factor graph ensembles

Recently, Vontobel showed the relationship between Bethe free energy and annealed free energy for protograph factor graph ensembles. In this paper, annealed free energy of any random regular factor graph ensembles are connected to Bethe free energy. The annealed free energy is expressed as the solution of maximization problem whose stationary condition coincides with equations of belief propagation since the contribution to partition function of particular type of variable and factor nodes has similar form of minus Bethe free energy. It gives simple derivation of quenched free energy by using the replica method. It implies equivalence of the replica and cavity methods for any random irregular factor graph ensembles. As consequence, it is shown that the replica symmetric solution and annealed free energy are equal for regular ensemble.

Loop Calculus For Nonbinary Alphabets Using Concepts From Information Geometry

The Bethe approximation is a well-known approximation of the partition function used in statistical physics. Recently, an equality relating the partition function and its Bethe approximation was obtained for graphical models with binary variables by Chertkov and Chernyak. In this equality, the multiplicative error in the Bethe approximation is represented as a weighted sum over all generalized loops in the graphical model. In this paper, the equality is generalized to graphical models with nonbinary alphabet using concepts from information geometry.

Holographic transformation, belief propagation and loop calculus for generalized probabilistic theories

The holographic transformation, belief propagation and loop calculus are generalized to problems in generalized probabilistic theories including quantum mechanics. In this work, the partition function of classical factor graph is represented by an inner product of two high-dimensional vectors both of which can be decomposed to tensor products of low-dimensional vectors. On the representation, the holographic transformation is clearly understood by using adjoint linear maps. Furthermore, on the formulation using inner product, the belief propagation is naturally defined from the derivation of the loop calculus formula. As a consequence, the holographic transformation, the belief propagation and the loop calculus are generalized to measurement problems in quantum mechanics and generalized probabilistic theories.