General Strong Polarization

by   Jarosław Błasiok, et al.

Arı kan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix M, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the polarization of an associated [0,1]-bounded martingale, namely its convergence in the limit to either 0 or 1 with probability 1. Arı kan showed appropriate polarization of the martingale associated with the matrix G_2 = ( < s m a l l m a t r i x >) to get capacity achieving codes. His analysis was later extended to all matrices M which satisfy an obvious necessary condition for polarization. While Arı kan's theorem does not guarantee that the codes achieve capacity at small blocklengths, it turns out that a "strong" analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with G_2 such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE IT '14]), thereby resolving a major theoretical challenge associated with the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are (in our view) also much simpler and modular. Key to our proof is a notion of local polarization that only depends on the evolution of the martingale in a single time step. Our result shows strong polarization over all prime fields and leads to efficient capacity-achieving source codes for compressing arbitrary i.i.d. sources, and capacity-achieving channel codes for arbitrary symmetric memoryless channels.


page 1

page 2

page 3

page 4


Arıkan meets Shannon: Polar codes with near-optimal convergence to channel capacity

Let W be a binary-input memoryless symmetric (BMS) channel with Shannon ...

Polar Codes with exponentially small error at finite block length

We show that the entire class of polar codes (up to a natural necessary ...

Capacity-achieving Polar-based LDGM Codes with Crowdsourcing Applications

In this paper we study codes with sparse generator matrices. More specif...

Capacity-achieving Polar-based Codes with Sparsity Constraints on the Generator Matrices

In this paper, we leverage polar codes and the well-established channel ...

Capacity-achieving Polar-based LDGM Codes

In this paper, we study codes with sparse generator matrices. More speci...

Reed-Muller codes polarize

Reed-Muller (RM) codes and polar codes are generated by the same matrix ...

Channel Polarization through the Lens of Blackwell Measures

Each memoryless binary-input channel (BIC) can be uniquely described by ...

Please sign up or login with your details

Forgot password? Click here to reset