{"date_published":"2021-09-01T00:00:00Z","intvolume":"        67","oa_version":"Preprint","title":"Binary linear codes with optimal scaling: Polar codes with large kernels","related_material":{"record":[{"status":"public","id":"6665","relation":"earlier_version"}]},"_id":"9002","date_updated":"2024-03-07T12:18:50Z","date_created":"2021-01-10T23:01:18Z","volume":67,"citation":{"ama":"Fazeli A, Hassani H, Mondelli M, Vardy A. Binary linear codes with optimal scaling: Polar codes with large kernels. <i>IEEE Transactions on Information Theory</i>. 2021;67(9):5693-5710. doi:<a href=\"https://doi.org/10.1109/TIT.2020.3038806\">10.1109/TIT.2020.3038806</a>","apa":"Fazeli, A., Hassani, H., Mondelli, M., &#38; Vardy, A. (2021). Binary linear codes with optimal scaling: Polar codes with large kernels. <i>IEEE Transactions on Information Theory</i>. IEEE. <a href=\"https://doi.org/10.1109/TIT.2020.3038806\">https://doi.org/10.1109/TIT.2020.3038806</a>","mla":"Fazeli, Arman, et al. “Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels.” <i>IEEE Transactions on Information Theory</i>, vol. 67, no. 9, IEEE, 2021, pp. 5693–710, doi:<a href=\"https://doi.org/10.1109/TIT.2020.3038806\">10.1109/TIT.2020.3038806</a>.","ieee":"A. Fazeli, H. Hassani, M. Mondelli, and A. Vardy, “Binary linear codes with optimal scaling: Polar codes with large kernels,” <i>IEEE Transactions on Information Theory</i>, vol. 67, no. 9. IEEE, pp. 5693–5710, 2021.","chicago":"Fazeli, Arman, Hamed Hassani, Marco Mondelli, and Alexander Vardy. “Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels.” <i>IEEE Transactions on Information Theory</i>. IEEE, 2021. <a href=\"https://doi.org/10.1109/TIT.2020.3038806\">https://doi.org/10.1109/TIT.2020.3038806</a>.","short":"A. Fazeli, H. Hassani, M. Mondelli, A. Vardy, IEEE Transactions on Information Theory 67 (2021) 5693–5710.","ista":"Fazeli A, Hassani H, Mondelli M, Vardy A. 2021. Binary linear codes with optimal scaling: Polar codes with large kernels. IEEE Transactions on Information Theory. 67(9), 5693–5710."},"publication":"IEEE Transactions on Information Theory","abstract":[{"lang":"eng","text":" 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 ε>0 of capacity, the code length n often scales as O(1/εμ), where the constant μ is called the scaling exponent. It is known that the optimal scaling exponent is μ=2, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the 2×2 kernel) on the BEC is μ=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×ℓ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent μ(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 μ(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(nlogn)."}],"month":"09","article_type":"original","external_id":{"arxiv":["1711.01339"]},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"MaMo"}],"publication_status":"published","publisher":"IEEE","publication_identifier":{"eissn":["1557-9654"],"issn":["0018-9448"]},"day":"01","author":[{"last_name":"Fazeli","full_name":"Fazeli, Arman","first_name":"Arman"},{"last_name":"Hassani","first_name":"Hamed","full_name":"Hassani, Hamed"},{"last_name":"Mondelli","full_name":"Mondelli, Marco","first_name":"Marco","orcid":"0000-0002-3242-7020","id":"27EB676C-8706-11E9-9510-7717E6697425"},{"last_name":"Vardy","full_name":"Vardy, Alexander","first_name":"Alexander"}],"page":"5693-5710","quality_controlled":"1","type":"journal_article","language":[{"iso":"eng"}],"article_processing_charge":"No","issue":"9","year":"2021","scopus_import":"1","doi":"10.1109/TIT.2020.3038806","status":"public"}