Polar-like Codes and Asymptotic Tradeoff among Block Length, Code Rate, and Error Probability
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A general framework is proposed that includes polar codes over arbitrary channels with arbitrary kernels. The 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). CapacityQuantitatively, if P=(-N^β^*) is possible for some β^* and if R=-N^1/μ^* is possible for some 1/μ^*, then (P,R)=((-N^β'),-N^-1/μ') is possible for some pair (β',1/μ') determined by β^*, 1/μ^*, 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 Arıkan's codes over BEC; [arXiv:1706.02458] analyzing Arıkan's 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 N N over BEC. In particular, (P,R)=((-N^.33),-N^-.33) is possible (Corollary 10). In fact, all points in this triangle are possible (β',1/μ')-pairs. enclose4̊e̊m̊1̊e̊m̊1̊e̊m̊^̊(0,1/2)_(0,0)left,bottom,downdiagonalstrike_̊(1,0)
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