The crossover list-size conjecture for polar codes

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Consider two polar codes AA and BB with the same code length nn and message length kk. Let FERSC(⋅){\rm FER_{SC}}(\cdot) and FERML(⋅){\rm FER_{ML}}(\cdot) denote their frame-error rates under successive-cancellation and maximum-likelihood decoding, respectively. Crossover list-size conjecture. If

FERSC(A)<FERSC(B),FERML(A)>FERML(B),{\rm FER_{SC}}(A) < {\rm FER_{SC}}(B),\qquad {\rm FER_{ML}}(A) > {\rm FER_{ML}}(B),

then there is a list size L′L^\prime such that code AA outperforms code BB for L<L′L<L^\prime, while code BB performs better for L>L′L>L^\prime at high signal-to-noise ratio. The paper gives an example using (128,64)(128,64) codes.

References

Primary source

Peihong Yuan, Tobias Prinz, Georg Böcherer, Onurcan İşcan, Ronald Böhnke and Wen Xu, “Polar Code Construction for List Decoding”, arXiv:1707.09753 (2018).

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