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Distributed Arithmetic Coding (DAC) is a practical realization of Slepian-Wolf coding that partitions source space into cosets. Coset Cardinality Spectrum (CCS) is an important property of DAC that was defined in our previous work. In this paper, we give two applications of CCS. First, we find that DAC bitstream is not compact. The rate loss of DAC bitstream is caused by two factors: unequal coset partitioning and bit indivisibility. It is proved that as code length goes to infinity, the expected value of bit-indivisibility rate loss will tend to 0.5 for any irrational Rate Change Step (RCS), where the RCS refers to the rate change when one source bit is flipped. With the help of CCS, the bit-indivisibility rate loss can be compensated to some extend. Especially, for any irrational RCS, as code length goes to infinity, the expected value of the remaining bit-indivisibility rate loss after compensation will tend to about 0.47. The second application of CCS is DAC decoder design. We derive the formula of path metric and find that in the original paper on DAC, the intuitive formula of path metric is not correct. The backward-replacing algorithm is proposed to make full use of memory. Experimental results confirm the correctness of theoretical analyses. [ABSTRACT FROM AUTHOR]

Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences (i.e., sequences with terms in {−1, 1}). If $f$ is a nonzero sequence, then its autocorrelation demerit factor, $\text {ADF}(f)$ , is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing $f$ to have unit Euclidean norm. If $(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demerit factor, $\text {CDF}(f,g)$ , is the sum of the squared magnitudes of the aperiodic crosscorrelation values over all shifts for the sequences obtained by normalizing both $f$ and $g$ to have unit Euclidean norm. Pursley and Sarwate showed that for binary sequences, the sum of $\text {CDF}(f,g)$ and the geometric mean of $\text {ADF}(f)$ and $\text {ADF}{(g)}$ must be at least 1. For randomly selected pairs of long binary sequences, this quantity is typically around 2. In this paper, we show that Pursley and Sarwate’s bound is met for binary sequences precisely when $(f,g)$ is a Golay complementary pair. We also prove a generalization of this result for sequences whose terms are arbitrary complex numbers. We investigate constructions that produce infinite families of Golay complementary pairs, and compute the asymptotic values of autocorrelation and crosscorrelation demerit factors for such families. [ABSTRACT FROM AUTHOR]

This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any $n$ -tuple occurs at most once in a period in either direction. As has been previously described, such sequences have potential applications in automatic position-location systems, where the sequence is encoded onto a surface and a reader needs only examine $n$ consecutive encoded bits to determine its location and orientation on the surface. The only previously described method of construction (due to Dai et al.) is somewhat complex, whereas the new techniques are simple to both describe and implement. The methods of construction cover both the standard ‘infinite periodic’ case, and also the aperiodic, finite sequence, case. Both the new methods build on the Lempel homomorphism, first introduced as a means of recursively generating de Bruijn sequences. [ABSTRACT FROM AUTHOR]

Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field $\mathbb {F}_{2^{n}}$ for a positive integer $n$. It is often overlooked in this community that all $2^{n}+1$ rational places (including “the place at infinity”) of the rational function field over $\mathbb {F}_{2^{n}}$ form a cyclic structure under an automorphism of order $2^{n}+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $2^{n}+1$ via cyclotomic function fields over $\mathbb {F}_{2^{n}}$. Each family of our sequences has size $2^{n}-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor $. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $2^{n}+1$. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^{n}-1$ for even $n$ , our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller. [ABSTRACT FROM AUTHOR]

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained by using the methods of multivariate analytic combinatorics, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion— $q$ -ary sequences with fixed Manhattan weight—is briefly discussed, and an application in coding for molecular timing channels is illustrated. [ABSTRACT FROM AUTHOR]

Given a binary sequence with length $n$ , determining its minimal rational fraction representation (MRFR) has important applications in the design and cryptanalysis of stream ciphers. There are many studies of this problem since Klapper and Goresky first introduced an adaptive rational approximation algorithm with a time complexity of $O(n^{2}\log n\log \log n)$. In this paper, we revisit this problem by considering both adaptive and non-adaptive efficient algorithms. Compared with the state-of-art methods, we make several contributions to the problem of finding MRFR. Firstly, we find a general and precise recursive relationship between the minimal bases for two adjacent lattices generated by successive truncation sequences. This enables us to improve the currently fastest adaptive algorithm proposed by Li et al.. Secondly, by optimizing a time-consuming step of the well-known Lagrange reduction algorithm for 2-dimensional lattices, we obtain a non-adaptive, and yet practically faster MRFR-solving algorithm named global Euclidean algorithm. Thirdly, we identify theoretical flaws on some non-adaptive methods in the literature by counter-examples and correct the problems by designing modified Euclidean algorithm named partial Euclidean algorithm. Meanwhile, we further reduce the time complexity of existing algorithm from $O(n^{2})$ to $O(n\log ^{2}n\log \log n)$ by invoking the half-gcd algorithm. We also conduct a comprehensive experimental comparative analysis on the above algorithms to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]