Lossless quantum data compression and variable-length coding

In order to compress quantum messages without loss of information it is necessary to allow the length of the encoded messages to vary. We develop a general framework for variable-length quantum messages in close analogy to the classical case and show that lossless compression is only possible if the message to be compressed is known to the sender. The lossless compression of an ensemble of messages is bounded from below by its von-Neumann entropy. We show that it is possible to reduce the number of qbits passing through a quantum channel even below the von-Neumann entropy by adding a classical side-channel. We give an explicit communication protocol that realizes lossless and instantaneous quantum data compression and apply it to a simple example. This protocol can be used for both online quantum communication and storage of quantum data.Comment: 16 pages, 5 figure

Comma-free Codes Over Finite Alphabets

Comma-free codes have been widely studied in the last sixty years, from points of view as diverse as biology, information theory and combinatorics. We develop new methods to study comma-free codes achieving the maximum size, given the cardinality of the alphabet and the length of the words. Specifically, we are interested in counting the number of such codes. We provide (two different proofs for) a closed-formula. The approach introduced is further developed to tackle well-known sub-families of comma-free codes, such as self-complementary and (generalisations of) non-overlapping codes. We also study codes that are not contained in strictly larger ones. For instance, we determine the maximal size of self-complementary comma-free codes and the number of codes reaching the bound. We provide a characterisation of-letter non-overlapping codes (over an alphabet of cardinality n), which allows us to devise the number of such codes that are not contained in any strictly larger one. Our approach mixes combinatorial and graph-theoretical arguments

MOCZ for Blind Short-Packet Communication: Practical Aspects

We investigate practical aspects of a recently introduced blind (noncoherent) communication scheme, called modulation on conjugate-reciprocal zeros (MOCZ). MOCZ is suitable for a reliable transmission of sporadic and short-packets at ultra-low latency and high spectral efficiency via unknown multipath channels, which are assumed to be static over the receive duration of one packet. The information is modulated on the zeros of the transmitted discrete-time baseband signal’s z− transform. Because of ubiquitous impairments between the transmitter and receiver clocks, a carrier frequency offset occurs after down-conversion to the baseband. This results in a common rotation of the zeros. To identify fractional rotations of the base angle in the zero-pattern, we propose an oversampled direct zero-testing decoder to identify the most likely one. Integer rotations correspond to cyclic shifts of the binary message, which we determine by cyclically permutable codes (CPC). Additionally, the embedding of CPCs into cyclic codes, enables additive error-correction which reduces the bit-error-rate tremendously. Furthermore, we exploit the trident structure in the signal’s autocorrelation for an energy based detector to estimate timing offsets and the effective channel delay spread. We finally demonstrate how this joint data and channel estimation can be largely improved by receive antenna diversity at low SNR

Synchronization with permutation codes and Reed-Solomon codes

D.Ing. (Electrical And Electronic Engineering)We address the issue of synchronization, using sync-words (or markers), for encoded data. We focus on data that is encoded using permutation codes or Reed-Solomon codes. For each type of code (permutation code and Reed-Solomon code) we give a synchronization procedure or algorithm such that synchronization is improved compared to when the procedure is not employed. The gure of merit for judging the performance is probability of synchronization (acquisition). The word acquisition is used to indicate that a sync-word is acquired or found in the right place in a frame. A new synchronization procedure for permutation codes is presented. This procedure is about nding sync-words that can be used speci cally with permutation codes, such that acceptable synchronization performance is possible even under channels with frequency selective fading/jamming, such as the power line communication channel. Our new procedure is tested with permutation codes known as distance-preserving mappings (DPMs). DPMs were chosen because they have de ned encoding and decoding procedures. Another new procedure for avoiding symbols in Reed-Solomon codes is presented. We call the procedure symbol avoidance. The symbol avoidance procedure is then used to improve the synchronization performance of Reed-Solomon codes, where known binary sync-words are used for synchronization. We give performance comparison results, in terms of probability of synchronization, where we compare Reed-Solomon with and without symbol avoidance applied

Some Single and Combined Operations on Formal Languages: Algebraic Properties and Complexity

In this thesis, we consider several research questions related to language operations in the following areas of automata and formal language theory: reversibility of operations, generalizations of (comma-free) codes, generalizations of basic operations, language equations, and state complexity. Motivated by cryptography applications, we investigate several reversibility questions with respect to the parallel insertion and deletion operations. Among the results we obtained, the following result is of particular interest. For languages L1, L2 ⊆ Σ∗, if L2 satisfies the condition L2ΣL2 ∩ Σ+L2Σ+ = ∅, then any language L1 can be recovered after first parallel-inserting L2 into L1 and then parallel-deleting L2 from the result. This property reminds us of the definition of comma-free codes. Following this observation, we define the notions of comma codes and k-comma codes, and then generalize them to comma intercodes and k-comma intercodes, respectively. Besides proving all these new codes are indeed codes, we obtain some interesting properties, as well as several hierarchical results among the families of the new codes and some existing codes such as comma-free codes, infix codes, and bifix codes. Another topic considered in this thesis are some natural generalizations of basic language operations. We introduce block insertion on trajectories and block deletion on trajectories, which properly generalize several sequential as well as parallel binary language operations such as catenation, sequential insertion, k-insertion, parallel insertion, quotient, sequential deletion, k-deletion, etc. We obtain several closure properties of the families of regular and context-free languages under the new operations by using some relationships between these new operations and shuffle and deletion on trajectories. Also, we obtain several decidability results of language equation problems with respect to the new operations. Lastly, we study the state complexity of the following combined operations: L1L2∗, L1L2R, L1(L2 ∩ L3), L1(L2 ∪ L3), (L1L2)R, L1∗L2, L1RL2, (L1 ∩ L2)L3, (L1 ∪ L2)L3, L1L2 ∩ L3, and L1L2 ∪ L3 for regular languages L1, L2, and L3. These are all the combinations of two basic operations whose state complexities have not been studied in the literature

On the Vocabulary of Grammar-Based Codes and the Logical Consistency of Texts

The article presents a new interpretation for Zipf's law in natural language which relies on two areas of information theory. We reformulate the problem of grammar-based compression and investigate properties of strongly nonergodic stationary processes. The motivation for the joint discussion is to prove a proposition with a simple informal statement: If an $n$-letter long text describes $n^\beta$ independent facts in a random but consistent way then the text contains at least $n^\beta/\log n$ different words. In the formal statement, two specific postulates are adopted. Firstly, the words are understood as the nonterminal symbols of the shortest grammar-based encoding of the text. Secondly, the texts are assumed to be emitted by a nonergodic source, with the described facts being binary IID variables that are asymptotically predictable in a shift-invariant way. The proof of the formal proposition applies several new tools. These are: a construction of universal grammar-based codes for which the differences of code lengths can be bounded easily, ergodic decomposition theorems for mutual information between the past and future of a stationary process, and a lemma that bounds differences of a sublinear function. The linguistic relevance of presented modeling assumptions, theorems, definitions, and examples is discussed in parallel.While searching for concrete processes to which our proposition can be applied, we introduce several instances of strongly nonergodic processes. In particular, we define the subclass of accessible description processes, which formalizes the notion of texts that describe facts in a self-contained way