We present a method to construct low-density parity-check (LDPC) codes with low error floors on the binary symmetric channel. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the trapping set ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum-product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four cycles.

/pdf/on-the-construction-of-structured-ldpc-codes-free-of-small-2g9tsqkcvy.pdf

On the Construction of Structured LDPC Codes Free of Small Trapping Sets

Quasi-cyclic LDPC codes are the most promising class of structured LDPC codes due to their ease of implementation and excellent performance over noisy channels when decoded with message-passing algorithms as extensive simulation studies have shown. In this paper, an approach for constructing quasi-cyclic LDPC codes based on Latin squares over finite fields is presented. By analyzing the parity-check matrices of these codes, combinatorial expressions for their ranks and dimensions are derived. Experimental results show that, with iterative decoding algorithms, the constructed codes perform very well over the AWGN and the binary erasure channels.

Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper, we draw on decomposition methods from optimization theory, specifically the alternating direction method of multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a “two-slice” characterization of the parity polytope, the polytope formed by taking the convex hull of all codewords of the single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by the ADMM decoder and its solution allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher SNR than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using the ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.

Decomposition Methods for Large Scale LP Decoding

岡山大学の仲田と申します.このたびは,寄稿の機会を頂 きありがとうございます.まず,自己紹介がてら,私が ISIT へ参加することになった経緯を書きたいと思います. 私の専門はWeyl 群やルート系が関わる表現論や組み合わ せ論です.その関係で,かつてから組み合わせ論サマース クールをはじめ千葉大学の萩原学氏と親交があったのです が,2016 年 12 月に沖縄のコーヒーショップで萩原氏か ら Varshmov-Tenengol’ts 符号が 1 bit 削除誤り訂正可 能であることを B型Weyl 群の言葉で記述するという着想 を伺いました.このとき,得もいわれぬ不思議さを感じたの が鮮明に記憶に残っております.B 型 Weyl が符号空間 ¥{0, 1 ¥}̂n と相性がよい,というのは今となれば当たり 前のことなのですが,当時の私にはよく知っていたと思い込 んでいた B 型Weyl 群に 1 bit 削除誤り訂正というまだ知 らない性質があったことも驚きでしたし,とても魅力を感じ ました.これをきっかけに 2018 年度から萩原氏と挿入削 除誤り訂正符号とWeyl 群のminuscule 元の関連性につい て研究を本格的に始め,私が ISIT への参加するようになっ たのも2018年が最初で,これ以降,毎年 ISIT へ参加する ようになりました.とはいえ,符号理論は私からすれば不慣 れな未知の分野で,一般常識も覚束ないままの研究でしたの で,ISIT へ submit するたびに reject されておりました. 2020 年頃からようやく勘所が分かってきたようで,今年 初めて,等しい複数 bit(複数の0か複数の1)を削除する 誤りを訂正できる完全符号の構成について念願の発表の機会 を頂きました. ISIT は,私が普段参加している研究集会よりもはるかに 大規模で,2018 年(アメリカ:コロラド)に初めて参加 したときは衝撃を受けました.非常に多くの参加者が,非常 に多くのパラレルセッションで発表しており,一体この人た ちはどういうネットワークになっているのだろう,と疑問が 一杯でした.また,研究領域の多様性にもただただ驚くばか りでした.ですので,このときはとにかく自分の研究に関係 のありそうな話題を時間が合えば聞きに行く,といった感じ でした.2020 年と 2021 年はオンライン開催だったの で,参加者の数を肌で感じることができず,その点はとても 残念でした.できれば,せっかくの初めての発表だったので 現地で人の前で話したいと思いました.とはいえ,動画によ る発表なので,幾つものセッションを時間帯を気にせず見る ことができるのはオンラインのよい点ですね.大学の仕事の 合間にあれこれ発表を見に行っていました.発表用の動画を 作るのもよい経験になりました.まず,動画作成に使う編集 ソフトを何にするかで試行錯誤するところから始まり,編集 し始めると無料の編集回数を超えてしまって,途中で編集ソ フトを切り替える,なんてことをやっていたので,私の発表 用動画はつぎはぎだらけでした.ほかの発表者の動画を見る と,内容もさることながら,ただただ皆構成や編集が上手だ なぁと感心するばかりです. 今回,ISIT へ参加・発表を通じて様々な経験をすること ができました.このことは私にとって大いに有意義でした. 願わくは,今度は対面で発表の機会を得たいですね.

IEEE international symposium on information theory

In this paper, we study the cycle structure of quasi-cyclic (QC) low-density parity-check (LDPC) codes with the goal of obtaining the shortest code with a given degree distribution and girth. We focus on QC-LDPC codes, whose Tanner graphs are cyclic liftings of fully connected base graphs of size $3 \times n, n \geq 4$ , and obtain minimal lifting degrees that result in girths 6 and 8. This is performed through an efficient exhaustive search, and as a result, we also find all the possible non-isomorphic codes with the same minimum block length, girth, and degree distribution. The exhaustive search, which is ordinarily a formidable task, is made possible by pruning the search space of many codes that are isomorphic to those previously examined in the search process. Many of the pruning techniques proposed in this paper are also applicable to QC-LDPC codes with base graphs other than the $3 \times n$ fully connected ones discussed here, as well as to codes with a larger girth. To further demonstrate the effectiveness of the pruning techniques, we use them to search for QC-LDPC codes with girths 10 and 12, and find a number of such codes that have a shorter block length compared with the best known similar codes in the literature. In addition, motivated by the exhaustive search results, we tighten the lower bound on the block length of QC-LDPC codes of girth 6 constructed from fully connected $3 \times n$ base graphs, and construct codes that achieve the lower bound for an arbitrary value of $n \geq 4$ .

Efficient Search of Girth-Optimal QC-LDPC Codes

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio in the case of communication over an additive white Gaussian noise channel.

Shortened Array Codes of Large Girth

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio (SNR) in the case of communication over an additive white Gaussian noise (AWGN) channel

pdf/shortened-array-codes-of-large-girth-2jtu72kv4d.pdf

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Shortened Array Codes of Large Girth

Shortened Array Codes of Large Girth