Binary Linear Codes With Optimal Scaling: Polar Codes With Large Kernels

被引:12
|
作者
Fazeli, Arman [1 ]
Hassani, Hamed [2 ]
Mondelli, Marco [3 ]
Vardy, Alexander [1 ,4 ]
机构
[1] Univ Calif San Diego, Dept Elect & Comp Engn, San Diego, CA 92093 USA
[2] Univ Penn, Dept Elect & Syst Engn, Philadelphia, PA 19104 USA
[3] Inst Sci & Technol Austria IST Austria, A-3400 Klosterneuburg, Austria
[4] Univ Calif San Diego, Dept Comp Sci & Engn, San Diego, CA 92093 USA
基金
美国国家科学基金会; 瑞士国家科学基金会;
关键词
Kernel; Complexity theory; Polar codes; Maximum likelihood decoding; Parity check codes; Channel coding; Capacity planning; Error-correcting codes; polar codes; polarization kernels; quasi-linear complexity; scaling exponent; REED-MULLER CODES; ACHIEVE CAPACITY; FINITE; EXPONENT; INEQUALITIES; POLARIZATION; PROBABILITY; BOUNDS;
D O I
10.1109/TIT.2020.3038806
中图分类号
TP [自动化技术、计算机技术];
学科分类号
0812 ;
摘要
We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within epsilon > 0 of capacity, the code length n often scales as O(1/epsilon(mu)), where the constant mu is called the scaling exponent. It is known that the optimal scaling exponent is mu = 2, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the 2 x 2 kernel) on the BEC is mu = 3.63. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist l x l binary kernels, such that polar codes constructed from these kernels achieve scaling exponent mu(l) that tends to the optimal value of 2 as l grows. We furthermore characterize precisely how large l needs to be as a function of the gap between mu(l) and 2. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity O(n) and encoding/decoding complexity O(n log n).
引用
收藏
页码:5693 / 5710
页数:18
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