Binary balanced codes approaching capacity

Abstract: In this paper, the construction of binary balanced codes is revisited. Binary balanced codes refer to sets of bipolar codewords where the number of “1”s in each codeword equals that of “0”s. The first algorithm for balancing codes was proposed by Knuth in 1986; however, its redundancy is almost two times larger than that of the full set of balanced codewords. We will present an efficient and simple construction with a redundancy approaching the minimal achievable one

Coding against synchronisation and related errors

In this thesis, we study aspects of coding against synchronisation errors, such as deletions and replications, and related errors. Synchronisation errors are a source of fundamental open problems in information theory, because they introduce correlations between output symbols even when input symbols are independently distributed. We focus on random errors, and consider two complementary problems: We study the optimal rate of reliable information transmission through channels with synchronisation and related errors (the channel capacity). Unlike simpler error models, the capacity of such channels is unknown. We first consider the geometric sticky channel, which replicates input bits according to a geometric distribution. Previously, bounds on its capacity were known only via numerical methods, which do not aid our conceptual understanding of this quantity. We derive sharp analytical capacity upper bounds which approach, and sometimes surpass, numerical bounds. This opens the door to a mathematical treatment of its capacity. We consider also the geometric deletion channel, combining deletions and geometric replications. We derive analytical capacity upper bounds, and notably prove that the capacity is bounded away from the maximum when the deletion probability is small, meaning that this channel behaves differently than related well-studied channels in this regime. Finally, we adapt techniques developed to handle synchronisation errors to derive improved upper bounds and structural results on the capacity of the discrete-time Poisson channel, a model of optical communication. Motivated by portable DNA-based storage and trace reconstruction, we introduce and study the coded trace reconstruction problem, where the goal is to design efficiently encodable high-rate codes whose codewords can be efficiently reconstructed from few reads corrupted by deletions. Remarkably, we design such n-bit codes with rate 1-O(1/log n) that require exponentially fewer reads than average-case trace reconstruction algorithms.Open Acces

L'algorithme de Pacman pour la construction efficace des codes équilibrés

Un bloc de bits est équilibré s’il contient un nombre de bits à 0 égal à celui des bits à 1. Les codes équilibrés (Balanced Codes, BC) sont largement appliqués dans plusieurs domaines. Par exemple, ils sont utilisés pour réduire le bruit dans les systèmes VLSI (Tabor, 1990). Dans le domaine de la télécommunication, ils sont utilisés dans la synchronisation et la transmission des données par fibre optique (Bergmann et al., 1986). Leur utilisation dans le domaine de l’identification par radiofréquence (RFID) permet d’augmenter les taux de transfert de données via les canaux RFID (Durgin, 2015). Étant donnée leur importance, plusieurs travaux de recherche ont été menés pour optimiser leur construction. Knuth fut le premier à trouver une méthode simple et rapide pour l’élaboration des codes équilibrés (Knuth, 1986). Il a introduit un algorithme très simple pour générer les codes équilibrés sans l’utilisation des tables de correspondance. Cependant, cet algorithme ajoute presque le double de la redondance minimale nécessaire pour la création des codes équilibrés. Une partie de cette redondance est due à la multiplicité d’encodage (ME) de cet algorithme. Plusieurs chercheurs ont essayé de réduire la redondance de l’algorithme de Knuth (Immink et Weber, 2009a, 2010; Immink et al., 2011; Al-Rababa’a et al., 2013). Dans les derniers travaux de Al-Rababa’a et al. (2013), les auteurs ont réussi à éliminer la redondance créée par ME et pourtant un écart par rapport au seuil minimal subsiste. Ce travail présente une alternative à l’algorithme de Knuth pour créer les codes équilibrés sans surplus de redondance. Nous proposons un algorithme nommé « Pacman » ¹ basé sur les permutations et les nombres entiers à précision limitée. En effet, le processus de codage de cet algorithme peut être assimilé à un Pacman qui consomme et produit des blocs d’informations d’une façon cyclique. Au cours de ce travail, nous allons montrer analytiquement et expérimentalement que la redondance introduite par notre technique est particulièrement faible, que les résultats sont nettement meilleurs que ceux des travaux antérieurs et que les complexités temporelle et spatiale utilisées sont linéaires. ¹. Inspiré de la marque de commerce PAC-MAN de l’entreprise BANDAI NAMCO.A block of m bits is said to be balanced if it contains an equal number of zeros and ones. Note that m has to be an even number. Balanced codes (BC) have applications in several domains. For example, they are used to reduce noise in VLSI systems (Tabor, 1990). In telecommunication, they are used in synchronization and data transmission by optical fibers (Bergmann et al., 1986). Their use in the field of radio frequency identification (RFID) can help to boost data transfer rate through RFID channels (Durgin, 2015). Given their importance, several research works have been carried out to optimize their construction. Knuth was the first to find a simple and fast method to create balanced codes (Knuth, 1986). He introduced a very simple algorithm to generate balanced codes without using lookup tables. However, Knuth’s balanced codes incur redundancy that is almost twice the one attested by the lower bound. A part of this redundancy is due to the multiplicity of encoding (ME) of this algorithm. Improvements to the Knuth’s algorithm are discussed in several research works (Immink et Weber, 2009a, 2010; Immink et al., 2011; Al-Rababa’a et al., 2013). In the last one (Al-Rababa’a et al., 2013), redundancy created by ME was eliminated and yet there is still some gap that needs to be closed. This work presents an alternative to Knuth’s algorithm for creating balanced codes without unwanted redundancy overhead. We propose an algorithm called "Pacman" ² that is based on permutations and limited-precision integers. Indeed, the coding process of this algorithm can be assimilated to a special Pacman that consumes and produces pills of information in a cyclical manner. In the presented work, we prove analytically and experimentally that our algorithm closes the mentioned redundancy gap while preserving a favorable compromise between calculation speed and memory consumption. ². Inspired by the trademark PAC-MAN of BANDAI NAMCO