Binary Alphabet Research Articles

On the generative power of regular pattern grammars

It is shown that for every recursively enumerable language L $$ \subseteq $$ ?* there exists a selective substitution grammar with a regular selector over a binary alphabet that generates L¢5, where ¢??. By requiring additional structural properties of the (already simple) selectors the language generating power is reduced in such a way that the resulting class lies strictly in between the family of EOL languages and the family of context-sensitive languages. For this class of languages some decision problems and normal forms are considered.

The shortest common supersequence problem over binary alphabet is NP-complete

We consider the complexity of the Shortest Common Supersequence (SCS) problem, i.e. the problem of finding for finite strings S 1, S 2,…, S u a shortest string S such that every S i can be obtained by deleting zero or more elements from S. The SCS problem is shown to be NP-complete for strings over an alphabet of size ⩾ 2.

Homomorphism equivalence on etol languages†

The following problems are shown to be decidable. Given an ETOL language L and two homomorphisms h 1 h 2is the length (Parikh vector) of h 1(x) and h 2(x) equal for each x in L? If Lis over a binary alphabet then we can also test whether h 1(x)= h 2(x) for each x in L.

A class of discrete signal-design problems in burst noise

The design of useful signaling alphabets in a burst-noise environment is treated as a discrete signal-design problem. A binary alphabet with minimum Gabor bandwidth is formulated as a discrete optimization problem to illustrate the design philosophy. The solution is obtained by using an eigenvalue approach. This technique is extended to the design of signals in a burst-noise environment where the burst is simulated by the removal of samples. Higher order alphabets are constructed for specific sample sizes and their performance evaluated.

Upper Bound on the Efficiency of dc-Constrained Codes

We derive the limiting efficiencies of dc-constrained codes. Given bounds on the running digital sum (RDS), the best possible coding efficiency η, for a K-ary transmission alphabet, is η = log <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> λ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> /log <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> K, where λ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> is the largest eigenvalue of a matrix which represents the transitions of the allowable states of RDS. Numerical results are presented for the three special cases of binary, ternary and quaternary alphabets.

The capacity of a channel with arbitrarily varying channel probability functions and binary output alphabet

Let X={1,..., a} be the “input alphabet” and Y={1,2} be the “output alphabet”. Let Xt=X and Yt=Y for t=1,2,..., Xn=\(\mathop \prod \limits_{t = 1}^n \)Xt and Yn=\(\mathop \prod \limits_{t = 1}^n \)Yt. Let S be any set, C=={w(·¦·¦)s)¦s∈S} be a set of (a×2) stochastic matrices w(·∥·¦s), and St=S, t=1,..., n. For every sn=(s1,...,sn)∈\(\mathop \prod \limits_{t = 1}^n \)St define P(·¦·¦sn)=\(\mathop \prod \limits_{t = 1}^n \)w(yt¦xt¦st) for every xn=x1, ⋯, xneXn and every yn=(y1, ⋯, yn)eYn. Consider the channel Cn={P(·¦·¦)sn)¦sn∈Sn} with matrices (·¦·¦s), varying arbitrarily from letter to letter. The authors determine the capacity of this channel when a) neither sender nor receiver knows sn, b) the sender knows sn, but the receiver does not, and c) the receiver knows sn, but the sender does not.

Data Transmission Error Probabilities in the Presence of Low-Frequency Removal and Noise

Upper bounds on error probability are derived for data transmission systems which are subjected to gaussian noise and to the removal of the low-frequency components of the signal. This error probability can be quite low for random data, even though the eye pattern is closed. Both standard format and partial response signaling are considered, as are binary and multilevel alphabets. Numerical results are given for a high-pass filter containing a single pole and for a cascade of several such identical filters.

Normal Approximations to the Error Rate for Hard-Limited Correlators

The approximation to tile normal law of the error transition probability for a correlation receiver preceded by a wideband hard limiter is reviewed. As is well known, the correlator output is asymptotically normal. This circumstance, however, is in general no justification for computing an error probability with the use of the complimentary error function. A binary message alphabet is assumed and the probability that the correlator output is positive (negative), given the transmission of a negative (positive) message bit, is examined with Chernoff bounding techniques as refined by Shannon and Gallagher. This method avoids the usual objections raised to the Edgeworth series when used to compute an error probability. Subject to restrictions of weak carrier-to-noise ratios at the limiter input and sufficient correlator processing gain that the error rate will be less than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> , the conclusion reached is that the error probability can indeed be computed, to within the accuracy of the mathematical idealization of the real channel, with the use of the complimentary error function; that is to say as if the correlator output were indeed normal.

The Structure and Properties of Binary Cyclic Alphabets

A code which is to be used for error control on a real data system is necessarily restricted by the nature of the transmitting equipment. These restrictions have no connection with the primary function of the code; indeed, they frequently eliminate most of the codes about which anything is known at present. For example, the code to be used for error control by detection and retransmission on the trunks between data switching centers is required to be a cyclic (or truncated cyclic) code with 744 information places and 20 parity check bits. The computational problem in this case is to locate those cyclic codes which have exactly 20 parity checks and a block length of 764 or greater, and pick the one which is best suited for error control over a particular channel. This paper outlines a procedure for attacking such problems. It describes how to locate the cyclic codes with a fixed block length and a fixed number of parity checks, if any such exist, and gives some methods of finding the number of code words of each weight in a particular code. If one knows the statistics of the channel it is then possible to estimate the error control properties of the code. The procedure depends on an analysis of the algebraic structure of cyclic codes, which is given in Section II of this paper. Section I contains step-by-step instructions with no mathematical justification. It is hoped that the theory presented in Section II may be useful in other applications.

Analysis of the relative information rates in digital systems using case coding

The method of increasing the rate of transmission of information in block-coded digital systems, known as case coding, is analysed. The rate increase may be positive or negative depending on the number of messages, their selection probabilities and the coded output-symbol alphabet. The treatment is not restricted to binary alphabets. Problems in the implementation of the coding system, such as storage capacity, are discussed and a workable method of encoding is presented.

A note off two binary signaling alphabets

A generalization of Hamming's single error correcting codes is given along with a simple maximum likelihood detection scheme. For small redundancy these alphabets are unexcelled. The Reed-Muller alphabets are described as parity check alphabets and a new detection scheme is presented for them.

A Class of Binary Signaling Alphabets

A class of binary signaling alphabets called “group alphabets” is described. The alphabets are generalizations of Hamming's error correcting codes and possess the following special features: (1) all letters are treated alike in transmission; (2) the encoding is simple to instrument; (3) maximum likelihood detection is relatively simple to instrument; and (4) in certain practical cases there exist no better alphabets. A compilation is given of group alphabets of length equal to or less than 10 binary digits.

Multi de Bruijn sequences

We generalize the notion of a de Bruijn sequence to a "multi de Bruijn sequence": a cyclic or linear sequence that contains every k-mer over an alphabet of size q exactly m times. For example, over the binary alphabet {0,1}, the cyclic sequence (00010111) and the linear sequence 000101110 each contain two instances of each 2-mer 00,01,10,11. We derive formulas for the number of such sequences. The formulas and derivation generalize classical de Bruijn sequences (the case m=1). We also determine the number of multisets of aperiodic cyclic sequences containing every k-mer exactly m times; for example, the pair of cyclic sequences (00011)(011) contains two instances of each 2-mer listed above. This uses an extension of the Burrows-Wheeler Transform due to Mantaci et al, and generalizes a result by Higgins for the case m=1.