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

• Published in: IEEE transactions on information theory Vol. 67; no. 9; pp. 5693 - 5710
• Main Authors: Fazeli, Arman, Hassani, Hamed, Mondelli, Marco, Vardy, Alexander
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.09.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Binary codes; Capacity planning; Channel coding; Codes; Communication; Complexity; Complexity theory; Encoding-Decoding; Error-correcting codes; Kernel; Kernels; Linear codes; Maximum likelihood decoding; Parity check codes; Polar codes; polarization kernels; quasi-linear complexity; Scaling; scaling exponent
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2020.3038806

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 <inline-formula> <tex-math notation="LaTeX">\varepsilon > 0 </tex-math></inline-formula> of capacity, the code length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> often scales as <inline-formula> <tex-math notation="LaTeX">O(1/\varepsilon ^{\mu }) </tex-math></inline-formula>, where the constant <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula> is called the scaling exponent . It is known that the optimal scaling exponent is <inline-formula> <tex-math notation="LaTeX">\mu =2 </tex-math></inline-formula>, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the <inline-formula> <tex-math notation="LaTeX">2\times 2 </tex-math></inline-formula> kernel) on the BEC is <inline-formula> <tex-math notation="LaTeX">\mu =3.63 </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">\ell \times \ell </tex-math></inline-formula> binary kernels, such that polar codes constructed from these kernels achieve scaling exponent <inline-formula> <tex-math notation="LaTeX">\mu (\ell) </tex-math></inline-formula> that tends to the optimal value of 2 as <inline-formula> <tex-math notation="LaTeX">\ell </tex-math></inline-formula> grows. We furthermore characterize precisely how large <inline-formula> <tex-math notation="LaTeX">\ell </tex-math></inline-formula> needs to be as a function of the gap between <inline-formula> <tex-math notation="LaTeX">\mu (\ell) </tex-math></inline-formula> and 2. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity <inline-formula> <tex-math notation="LaTeX">O(n) </tex-math></inline-formula> and encoding/decoding complexity <inline-formula> <tex-math notation="LaTeX">O(n\log n) </tex-math></inline-formula>.