Constant Weight Codes Research Articles

Half Weight Block Codes for Optical Communications

This paper discusses the possibilities of using certain constant-weight block codes in optical data links and in optical ring networks. The codes that appear to be particulary attractive have exactly as many 1s as 0s in each code word, so there is no penalty for ac coupling. Codes can be simple, such as the Manchester code. Or they may be more complicated and support control channel signaling, error detection, and forward error correction. This paper bridges the gap that sometimes exists between the mathematically “ideal” block codes and the typical requirements of particular applications. It presents a variety of constant-weight codes of increasing length, including a code of length 16 that is being implemented for a very high-speed, ultra-reliable, ring data network.

Lexicographic codes: Error-correcting codes from game theory

Lexicographic codes, or lexicodes, are defined by various versions of the greedy algorithm. The theory of these codes is closely related to the theory of certain impartial games, which leads to a number of surprising properties. For example, lexicodes over an alphabet of size B=2^{a} are closed under addition, while if B = 2^{2^{a}} the lexicodes are closed under multiplication by scalars, where addition and multiplication are in the nim sense explained in the text. Hamming codes and the binary Golay codes are lexieodes. Remarkably simple constructions are given for the Steiner systems S(5,6,12) and S(5,8,24) . Several record-breaking constant weight codes are also constructed.

A modification of the Zinoviev lower bound for constant weight codes

In [3] Zinoviev presented a new method to get lower bounds for constant weight codes. In this note we show that a simple modification of the Zinoviev method gives further improvements.

Energy balance code for the simulation of D-T, D-3He and catalyzed D-D burning in high-field tokamaks of the OMITRON type

A zero-dimensional energy balance code is developed for the simulation of the approach to ignition of the D-T, D-3He and catalyzed D-D fuels (the last two fuels via an initial D-T burning trigger phase). With reference to a device of the OMITRON type, the requirements on linear dimensions, aspect ratio and average plasma «beta» are determined for the achievement of ignition in the indicated fuels according at least to extrapolations of the current understanding of plasma behaviour.

Line coding and intersymbol interference in local network transmission systems

Abstract The intersymbol interference experienced in digital transmission over local network cables is characterized by a simple empirical model of the cable pulse response. Using the model, it is shown that balanced linear binary codes, especially WAL 2, have sufficient self-equalization to give an open eye over the range of cable lengths for which crosstalk is not a problem. The multiple response codes (1, −1) and (1, 0, − 1) are shown not to confer any significant advantage over binary transmission in the absence of any other equalization. However, they do enable the length of a decision feedback equalizer used to reduce the intersymbol interference to be significantly shortened. This effect is quantified using an approximate formula for the bit error rate resulting from both the residual intersymbol interference and additive gaussian noise ; the validity of the formula is confirmed by simulations.

A note on the existence of perfect constant weight codes

By using an appropriate anticode it is shown that the Johnson graph J(n, ν) can contain an e-perfect subset only if ν ⩽ (n − 1)(2e + 1) e .

On group-theoretic codes for asymmetric channels

We study single error-correcting codes for the asymmetric channel with input and output alphabets being {0, 1,…, a — 1⩽ . From an abelian group G of order N with elements g 0 = 0, g 1 ,…, g N—1 , Constantin and Rao (1979, Inform. Contr. 40 , 20–36) define V g = {(b 1 ,b 2 ,...,b n−1 ) ∈ {O, 1,..., a − 1 } N−1 | ∑ N−1 i−1 b i g i 0 g} and show that V g correct single errors. We give explicit expressions for the size and weight distribution of these codes. We further give a short discussion of some constant weight codes obtained by a similar construction.

Some new upper bounds for constant weight codes (Corresp.)

The Johnson bound is used to obtain 12 new upper bounds for constant weight codes of length not exceeding 24.

A few new constant weight codes (Corresp.)

A method for searching for constant weight codes is described and used to construct 14 codes which are better than the known codes with the same parameters.

Lower bounds for constant weight codes

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(n,2\delta,w)</tex> denote the maximum number of codewords in any binary code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , constant weight <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> , and Hamming distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2\delta</tex> Several lower bounds for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(n,2\delta,w)</tex> are given. For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta</tex> fixed, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(n,2\delta,w) \geq n^{W-\delta+l}/w!</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(n,4,w)\sim n^{w-l}/w!</tex> as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \rightarrow \infty</tex> . In most cases these are better than the "Gilbert bound." Revised tables of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(n,2 \delta,w)</tex> are given in the range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \leq 24</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta \leq 5</tex> .

7B8B balanced code with simple error detecting capability

A description is given of a simple in-service error detection technique suitable for use at dependent repeaters. The technique is similar to even mark parity error detection but differs in that no special signals such as parity check digits are required. In fact the line coding terminal equipment remains unmodified by the inclusion of this error detection technique.

Design of efficient coding and modulation for a Rayleigh fading channel

The design of a coding/modulation structure for digital communications over a Rayleigh fading channel, the structure of the corresponding decoder, and the error rate performance of the resulting system are considered. Emphasis is on the use of constant weight codes for constructing equal energy waveforms for transmission over the channel. The performance gains that are achieved by the integrated coding/modulation approach relative to conventional methods for obtaining diversity are illustrated via some examples. Of special interest is the use of a concatenated coding technique for forming codes of large distance and hence high diversity. A new decoding algorithm is applied to enable efficient decoding of the concatenated code. An example is included that shows a performance increase of several dB resulting from concatenation.

Ionospheric heating by radio waves: Predictions for Arecibo and the satellite power station

The effect of resistive heating by radio waves on ionospheric temperatures, electron densities, and airglow emissions is examined by using numerical ionospheric structure and heat balance codes. Two cases are studied: (1) a 3‐GHz, 10‐GW microwave beam from a proposed satellite power station and (2) 1‐MW and 3‐MW beams of 15‐MHz radio waves launched by the Arecibo antenna. By intent, these two cases have similar intensities and geometries of resistive heating. The most dramatic heating effects are predicted to occur in the E region, where a thermal runaway will take place. The E region electron temperature will increase from 200°K to roughly 1000°K, and the E region electron density will increase by a factor of about 3. In the F region, where thermal conductivity plays an important role, temperature increases of 200°–500°K will appear along magnetic field lines passing through the radio wave beams. Enhanced emissions in airglow and molecular infrared lines will also occur. Radio wave heating, when combined with the diagnostic capabilities of the Arecibo incoherent scatter radar, will generate new opportunities to measure the rates of atomic physics processes and neutral atmosphere temperatures and composition at D and E region altitudes.

Upper bounds on the cardinality of a binary code with a given minimum distance

This paper obtains an upper bound on the cardinality of a binary code of length n and minimum distance d , which improves on the Elias bound exponentially in n . A new upper bound for the cardinality of a constant-weight binary code is also presented.

On the minimal redundancy of binary error-correcting codes

This paper obtains new lower bounds for the redundancy of both arbitrary and constant-weight binary codes with a fixed error-correcting capability.

Distance-2 Cyclic Chaining of Constant-Weight Codes

The cyclic distance-2 chaining of constant-weight codes has applications in A/ D conversions as well as in combinatorial problems involving the exhaustion of m-out-of-n combinations. It is shown in this paper that such a chaining can be obtained from the Gray code circuit and its transformations. Algorithms based on several theorems derived have been developed and programmed in APL.

Upper bounds for constant weight error correcting codes

In earlier papers the author has developed upper bounds for A( n, d), the maximum number of binary words of length n, each pair of words being at a Hamming distance of at least d apart. These sphere packing bounds for A( n, d) depend directly on a related bound for constant weight codes where each word has the same number of 1's. Improving bounds on constant weight codes would therefore improve the bounds on A( n, d). This was the motivation of the present study, but constant weight codes are interesting in their own right. By combining special techniques from several sources, a new upper bound for constant weight codes is developed which often gives significantly improved results.

A Survey of Coding Schemes for Transmission or Recording of Digital Data

In this survey we shall review coding techniques and results which pertain to such problems as reduction of dc wandering, suppression of intersymbol interference, and inclusion of selfclocking capability. These problems are of engineering interest in the transmission or recording of digital data. The topics to be discussed include: 1) dc free codes such as bipolar signals and feedback balanced codes, 2) correlative level codes and optimal decoding methods, 3) Fibonacci codes and run-length constraint codes, and 4) state-oriented codes.

Feedback Balanced Code for Multilevel PCM Transmission

This paper describes a new code format called the feedback balanced code (FBC) for multilevel pulse code modulation (PCM) transmission, which is easily implemented and is capable of efficient utilization of transmission cable. The FBC is a type of multilevel code which has no dc wander and is obtained by the polarity control of the output pulse sequence using negative feedback. The principle of the FBC and its static balance condition are discussed, and spectral analysis using transition probability is described. FBCs are shown to have spectral nulls at dc and at frequency f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> . Suppression of the low-frequency component is also represented by a transfer function of the FBC converter by employing an appropriate assumption of linearization. The suppression characteristic is much improved by using a double integration feedback network. A higher order FBC is also discussed, which has spectral nulls at frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_{o}/n</tex> and its integral multiples. The preceding theoretic approach was well verified through experiments.

Addendum: Linear Codes of Constant Weight

Addendum: Linear Codes of Constant Weight