Explicit Polar Codes with Small Scaling Exponent

Abstract

Polar coding gives rise to the first explicit family of codes that provably achieve capacity for a wide range of channels with efficient encoding and decoding. But how fast can polar coding approach capacity as a function of the code length? In finite-length analysis, the scaling between code length and the gap to capacity is usually measured in terms of the scaling exponent µ. It is well known that the optimal scaling exponent, achieved by random binary codes, is µ = 2. It is also well known that the scaling exponent of conventional polar codes on the binary erasure channel (BEC) is µ = 3.627, which falls far short of the optimal value. On the other hand, it was recently shown that polar codes derived from ℓ × ℓ binary polarization kernels approach the optimal scaling exponent µ = 2 on the BEC as ℓ → ∞, with high probability over a random choice of the kernel.Herein, we focus on explicit constructions of ℓ × ℓ binary kernels with small scaling exponent for ℓ 64. In particular, we exhibit a sequence of binary linear codes that approaches capacity on the BEC with quasi-linear complexity and scaling exponent µ < 3. To the best of our knowledge, such a sequence of codes was not previously known to exist. The principal challenges in establishing our results are twofold: how to construct such kernels and how to evaluate their scaling exponent.In a single polarization step, an ℓ × ℓ kernel K ℓ transforms an underlying BEC into ℓ bit-channels W 1 , W 2 , . . . , W ℓ . The erasure probabilities of W 1 , W 2 , . . . , W ℓ , known as the polarization behavior of K ℓ , determine the resulting scaling exponent µ(K ℓ ). We first introduce a class of self-dual binary kernels and prove that their polarization behavior satisfies a strong symmetry property. This reduces the problem of constructing K ℓ to that of producing a certain nested chain of only ℓ/2 self-orthogonal codes. We use nested cyclic codes, whose distance is as high as possible subject to the orthogonality constraint, to construct the kernels K 32 and K 64 . In order to evaluate the polarization behavior of K 32 and K 64 , two alternative trellis representations (which may be of independent interest) are proposed. Using the resulting trellises, we show that µ(K 32 ) = 3.122 and explicitly compute over half of the polarization-behavior coefficients for K 64 , at which point the complexity becomes prohibitive. To complete the computation, we introduce a Monte-Carlo interpolation method, which produces the estimate µ(K 64 ) ≃ 2.87. We augment this estimate with a rigorous proof that µ(K 64 ) < 2.97.

On the Polarizing Behavior and Scaling Exponent of Polar Codes with Product Kernels

“…It is already known that µ = 3.627 for conventional polar codes (Arikans T 2 kernel) over BEC [5]. Recently, a class of self-dual binary kernels were introduced in [6] in which large kernels of size 2 p were constructed with low µ and it was shown that µ = 3.122 for L = 32 and µ 2.87 for L = 64.…”

“…Kernel codes C i are defined as C i = g i+1 , g i+2 , · · · , g l for 0 i < l and C l = {0}. This kernel is said to be self-dual if C i = C ⊥ l−i for all 0 i l. Some of the properties of self dual kernel proved in [6] are: Property 1. ∀ w : E i,w + E l+1−i,l−w l w for i = 1, · · · , l.…”

“…When z is close to 0 in (5), the polynomial p i (z) is dominated by the first non-zero term E i,w z w (1 − z) (l−w) . From [6], we know that the first non-zero coefficients of p i (z) is E idi . For construction of self-dual kernel, we aim to maximize the partial distance to make p i (z) polarize to 0.…”

The Penalty in Scaling Exponent for Polar Codes Is Analytically Approximated by the Golden Ratio

“…A cellular automaton consists of a regular grid of discrete finite-state sites evolving according to some pre-defined parallel update rule with a discrete time variable t ∈ N. The celebrated Domany-Kinzel (DK) model [8] is a stochastic cellular automaton defined on a tilted square lattice 4 , as 3 Clearly the trivial recursion Zn → Z n+1 does not support the existence of the complementary polarization towards 1. This subtlety is addressed and circumvented in Section III-A.…”

“…For instance, their scaling exponent for the binary erasure channel (BEC) was numerically computed, based on a heuristic eigen-analysis of the polarization operator, to be about µ num ≃ 3.627 [4]. For such a case, lower and upper analytical bounds on the scaling exponent were derived, 1 A recent discussion on polar code constructions based on l × l binary polarization kernels with lower scaling exponents can be found in [3]. Fig.…”