Polar Codes and Polar Lattices for the Heegard–Berger Problem

Abstract

Explicit coding schemes are proposed to achieve the rate-distortion function of the Heegard-Berger problem using polar codes. Specifically, a nested polar code construction is employed to achieve the rate-distortion function for the doubly-symmetric binary sources when the side information may be absent. The nested structure contains two optimal polar codes for lossy source coding and channel coding, respectively. Moreover, a similar nested polar lattice construction is employed when the source and the side information are jointly Gaussian. The proposed polar lattice is constructed by nesting a quantization polar lattice and a capacity-achieving polar lattice for the additive white Gaussian noise channel. Index TermsHeegard-Berger Problem, source coding, lattices.Heegard-Berger problem, and prove that they achieve the HBRDF for doubly symmetric binary sources (DSBS). We consider the reconstruction of the source sequence at Decoder 1, i.e., the decoder without side information, denoted byX 1:N 1 and the original source sequence X 1:N as a combined source, and further combine this reconstructionX 1:N 1 with the original side information Y 1:N to obtain a combined side information. By this argument, we obtain another nested construction of polar codes, which achieves the HBRDF of the entire nondegenerate region. In addition, we present an explicit coding scheme by using two-level polar codes to achieve the HBRDF whose forward test channel may be asymmetric. Finally, we prove that polar codes achieve an exponentially decaying block error probability and excess distortion at both decoders for the binary Heegard-Berger problem.• We then consider the Gaussian Heegard-Berger problem, and propose a polar lattice construction that consists of two nested polar lattices, one of which is additive white Gaussian noise (AWGN) capacity-achieving while the other is Gaussian rate-distortion function achieving. This construction is similar to the one proposed for the Gaussian Wyner-Ziv problem in [6]. However, in the Heegard-Berger problem setting, we need to treat the difference between the original source and its reconstruction at Decoder 1 as a new source, and the difference between the original side information and the reconstruction at Decoder 1 as a new side information. As a result, we can obtain an optimal test channel that connects the new source with the new side information by using additive Gaussian noises. According to this test channel, we can further construct two nested polar lattices that achieve the Gaussian HBRDF of the entire non-degenerate region.

Security-based low-density parity check encoder for 5G communication

“…The earlier technique has numerous shortcomings, including greater power, space, and delay consumption. Hardware implementation of some functions can be challenging [4].…”