Fifth Generation (5G) New Radio (NR) Channel Codes Contenders Based on Field- Programmable Gate Arrays (FPGA): A Review Paper

ان الحاجة المتزايدة على الجودة، مثل السرعة العالية والتاخير المنخفض والتغطية الواسعة واستهلاك الطاقة والتكلفة والاتصالات الموثوقة في خدمات الهاتف المحمول والوسائط المتعددة ونقل البيانات تفرض استخدام المتطلبات التقنية المتقدمة في الجيل الخامس (5G) الإذاعة الجديدة (NR). واحدة من أهم الأجزاء في الطبقة المادية للجيل الجديد هي تقنية الترميز لتصحيح الأخطاء. هنالك ثلاثة اشكال مقترحة لتقنيات الترميز المخصصة لقنوات نقل البيانات وقنوات التحكم هي  الترميز التوربيني وفحص التكافؤ المنخفض الكثافة (LDPC) والرموز القطبية. يتم تقييم المنافسة بين هذه الانواع من حيث القدرة على تصحيح الأخطاء والتعقيد الحسابي والمرونة. التوازي والمرونة وسرعة المعالجة العالية لمصفوفة البوابة القابلة للبرمجة الميدانية (FPGA) تجعلها أفضل في النماذج الأولية وتنفيذ الرموز المختلفة. تقدم هذه الورقة دراسة استقصائية للبحوث الحالية التي تتعامل مع تصميم وحدة فك الترميز المستندة إلى FPGA المرتبطة برموز القناة المذكورة سابقًا.The increased demands for quality, like high throughput, low-latency, wide coverage, energy consumption, cost and reliable connections in mobile services, multimedia and data transmission impose the use of advance technical requirements for the next fifth-generation (5G) new radio (NR). One of the most crucial parts in the physical layer of the new generation is the error correction coding technique. Three schemes, namely; Turbo, low density parity check (LDPC), and polar codes are potentially ‎considered as the candidate codes for both data and control channels. The competition is evaluated in terms of error correction capability, computational complexity, and flexibility. The parallelism, flexibility and high processing speed of Field-Programmable Gate Array (FPGA) make it preferable in prototyping and implementation of different codes. This paper presents a survey on the current literatures that deals with FPGA-based decoder design associated with the previously mentioned channel codes

X-code: MDS array codes with optimal encoding

We present a new class of MDS (maximum distance separable) array codes of size n×n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns

Shortening array codes and the perfect 1-factorization conjecture

The existence of a perfect 1-factorization of the complete graph with n nodes, namely, K_n , for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations of K_(p+1) and K_2p , where p is an arbitrary prime number (p > 2) . It was shown in previous work that finding a perfect 1-factorization of K_n is related to a problem in coding, specifically, it can be reduced to constructing an MDS (Minimum Distance Separable), lowest density array code. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the K_(p+1) family of perfect 1-factorization from the K_2p family. Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related

M-spotty rosenbloom-tsfasman ağırlık sayacı için macwilliams özdeşlikleri

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Geniş bir kullanım alanına sahip olan hata kontrol kodları, son yıllarda bilgisayar ve iletişim sistemlerindeki güvenirliliği arttırma adına birinci derecede önem kazanmıştır. Bu bağlamda, bilgisayar hafıza sistemlerindeki güvenirliliği arttırmak için hata kontrol kodlarının bir sınıfı olan m-spotty parça hata kontrol kodları inşa edilmiştir. M-spotty parça hata kontrol kodları sayesinde yüksek yoğunluklu yarı iletken RAM yongaları güçlü elektromanyetik dalgalar, radyoaktif parçacıklar ya da kozmik parçacıklardan dolayı meydana gelen ardışık hatalar etkili bir şekilde tespit edilebilmekte ve ya düzeltilebilmektedir. İlk olarak, Hamming metriği kullanılarak m-spotty parça hata kontrol kodları inşa edilmiştir. Bu çalışma da ise m-spotty Rosenbloom-Tsfasman metriği adı verilen yeni bir metrik tanımlanarak m-spotty parça hata kontrol kodları ile bazı çalışmalar yapılmaktadır. Bu çalışmalar, m-spotty Rosenbloom-Tsfasman ağırlık sayaçları için MacWilliams özdeşliklerinin elde edilmesi ve matris kodları ile m-spotty Rosenbloom-Tsfasman ağırlığının ilişkilendirilmesi çalışmasıdır.Bu tez altı bölümden oluşmaktadır. Birinci bölümde, cebirsel yapıların özellikleri ile ilgili tanımlar ve teoremler, lineer kodların yapısı, m-spotty parça hata kontrol kodları ve MacWilliams özdeşliği hakkında bilgi verilmektedir.İkinci bölümde, sonlu cisimler üzerinde m-spotty parça hata kontrol kodları için yapılmış çalışmalar üzerinde durulmaktadır.Üçüncü ve dördüncü bölümlerde, yeni tanımlanan m-spotty Rosenbloom-Tsfasman metriğine göre farklı cebirsel yapılar ele alınarak, m-spotty Rosenbloom-Tsfasman ağırlık sayaçları için MacWilliams özdeşlikleri elde edilmektedir.Beşinci bölümde, m-spotty Rosenbloom-Tsfasman ağırlığı ile matris kodları ilişkilendirilmektedir.Altıncı ve son bölüm, sonuç ve öneriler kısmından oluşmuştur.Error control codes have extensively been applied to semiconductor memories using high density RAM chips with wide Input/Output data. High density semiconductor RAM chips with wide Input/Output, e.g., with 8-bit or 16-bit Input/Output data, have become popular in recent years. However, these semiconductor memories are highly vulnerable to multiple random bit errors when they are exposed to strong electromagnetic waves, radio active particles, or energetic cosmic particles. These multiple random bit errors, typically 2- or 3-bit errors, are usually confined to a byte region because RAM chips, each of which corresponds to a byte having length bits, are physically independent. These errors can be effectively corrected or detected by m-spotty byte error control codes. Firstly, m-spotty byte error control codes have been characterized by the m-spotty Hamming distance. This study defines a new distance function, called m-spotty Rosenbloom-Tsfasman distance, of the m-spotty byte error control codes. Morever, this study proposes a MacWilliams type identity for m-spotty Rosenbloom-Tsfasman weight enumerators via the new distance function, which is a metric. However, it is shown that there exists a relationship between array codes and the m-spotty Rosenbloom-Tsfasman weight.This thesis consists of six chapters. In the first chapter, some basic definitions and theorems related to algebraic structures, some information about linear codes, m-spotty byte error control codes and MacWilliams identity are given.The second chapter presents a survey of the m-spotty byte error control codes over finite fields.In the third and fourth chapters, the MacWilliams identity for m-spotty Rosenbloom-Tsfasman weight enumerators over some algebraic structures is obtained.In the fifth chapter, the array codes having the m-spotty Rosenbloom-Tsfasman weight are introduced.In the sixth and the last chapter, the conclusion and possible work are given

Low-density MDS codes and factors of complete graphs

We present a class of array code of size n×l, where l=2n or 2n+1, called B-Code. The distances of the B-Code and its dual are 3 and l-1, respectively. The B-Code and its dual are optimal in the sense that i) they are maximum-distance separable (MDS), ii) they have an optimal encoding property, i.e., the number of the parity bits that are affected by change of a single information bit is minimal, and iii) they have optimal length. Using a new graph description of the codes, we prove an equivalence relation between the construction of the B-Code (or its dual) and a combinatorial problem known as perfect one-factorization of complete graphs, thus obtaining constructions of two families of the B-Code and its dual, one of which is new. Efficient decoding algorithms are also given, both for erasure correcting and for error correcting. The existence of perfect one-factorizations for every complete graph with an even number of nodes is a 35 years long conjecture in graph theory. The construction of B-Codes of arbitrary odd length will provide an affirmative answer to the conjecture

Asymptotically MDS Array BP-XOR Codes

Belief propagation or message passing on binary erasure channels (BEC) is a low complexity decoding algorithm that allows the recovery of message symbols based on bipartite graph prunning process. Recently, array XOR codes have attracted attention for storage systems due to their burst error recovery performance and easy arithmetic based on Exclusive OR (XOR)-only logic operations. Array BP-XOR codes are a subclass of array XOR codes that can be decoded using BP under BEC. Requiring the capability of BP-decodability in addition to Maximum Distance Separability (MDS) constraint on the code construction process is observed to put an upper bound on the maximum achievable code block length, which leads to the code construction process to become a harder problem. In this study, we introduce asymptotically MDS array BP-XOR codes that are alternative to exact MDS array BP-XOR codes to pave the way for easier code constructions while keeping the decoding complexity low with an asymptotically vanishing coding overhead. We finally provide and analyze a simple code construction method that is based on discrete geometry to fulfill the requirements of the class of asymptotically MDS array BP-XOR codes.Comment: 8 pages, 4 figures, to be submitte