2015/01/11 by Marco Mondelli, Mondelli, Marco, S. Hamed Hassani +3 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · #Advanced biosensing and bioanalysis techniques #DNA and Biological Computing #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.1501.02444
openalex publication_date 2015/01/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Consider the transmission of a polar code of block length N and rate R\nover a binary memoryless symmetric channel W and let Pe be the block error\nprobability under successive cancellation decoding. In this paper, we develop\nnew bounds that characterize the relationship of the parameters R, N,\nPe, and the quality of the channel W quantified by its capacity I(W) and\nits Bhattacharyya parameter Z(W).\n In previous work, two main regimes were studied. In the error exponent\nregime, the channel W and the rate R<I(W) are fixed, and it was proved that\nthe error probability Pe scales roughly as 2-\√(N). In the scaling\nexponent approach, the channel W and the error probability Pe are fixed\nand it was proved that the gap to capacity I(W)-R scales as N-1/\μ.\nHere, \μ is called scaling exponent and this scaling exponent depends on the\nchannel W. A heuristic computation for the binary erasure channel (BEC) gives\n\μ=3.627 and it was shown that, for any channel W, 3.579 \≤ \μ \≤\n5.702.\n Our contributions are as follows. First, we provide the tighter upper bound\n\μ \≤ 4.714 valid for any W. With the same technique, we obtain \μ \≤\n3.639 for the case of the BEC, which approaches very closely its heuristically\nderived value. Second, we develop a trade-off between the gap to capacity\nI(W)-R and the error probability Pe as functions of the block length N.\nIn other words, we consider a moderate deviations regime in which we study how\nfast both quantities, as functions of the block length N, simultaneously go\nto 0. Third, we prove that polar codes are not affected by error floors. To\ndo so, we fix a polar code of block length N and rate R. Then, we vary the\nchannel W and we show that the error probability Pe scales as the\nBhattacharyya parameter Z(W) raised to a power that scales roughly like\n\√(N).\n