Paper Info
Title
Polar-like Codes and Asymptotic Tradeoff among Block Length, Code Rate, and Error Probability.
Abstract
A general framework is proposed that includes polar codes over arbitrary channels with arbitrary kernels. asymptotic tradeoff among block length $N$, code rate $R$, and error probability $P$ is analyzed. Given a tradeoff between $N,P$ and a tradeoff between $N,R$, we return an interpolating tradeoff among $N,R,P$ (Theorem 5). $defCapacity{text{Capacity}}$Quantitatively, if $P=exp(-N^{beta^*})$ is possible for some $beta^*$ and if $R=Capacity-N^{1/mu^*}$ is possible for some $1/mu^*$, then $(P,R)=(exp(-N^{betau0027}),Capacity-N^{-1/muu0027})$ is possible for some pair $(betau0027,1/muu0027)$ determined by $beta^*$, $1/mu^*$, and auxiliary information. In fancy words, an error exponent regime tradeoff plus a scaling exponent regime tradeoff implies a moderate deviations regime tradeoff. The current world records are: [arXiv:1304.4321][arXiv:1501.02444][arXiv:1806.02405] analyzing Ar{i}kanu0027s codes over BEC; [arXiv:1706.02458] analyzing Ar{i}kanu0027s codes over AWGN; and [arXiv:1802.02718][arXiv:1810.04298] analyzing general codes over general channels. An attempt is made to generalize all at once (Section IX). As a corollary, a grafted variant of polar coding almost catches up the code rate and error probability of random codes with complexity slightly larger than $Nlog N$ over BEC. In particular, $(P,R)=(exp(-N^{.33}),Capacity-N^{-.33})$ is possible (Corollary 10). In fact, all points in this triangle are possible $(betau0027,1/muu0027)$-pairs. $$ require{enclose} defr{phantom{Rule{4em}{1em}{1em}}} enclose{}r^llap{(0,1/2)}_llap{(0,0)} enclose{left,bottom,downdiagonalstrike}r_rlap{(1,0)} enclose{}r $$
Year
Venue
DocType
2018
arXiv: Information Theory
Journal
Volume
Citations 
PageRank 
abs/1812.08112
0
0.34
References 
Authors
0
2
Name
Order
Citations
PageRank
Hsin-Po Wang101.35
Iwan M. Duursma227926.85