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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

The weight distribution and randomness of linear codes

Finding the weight distributions of block codes is a problem of theoretical and practical interest. Yet the weight distributions of most block codes are still unknown except for a few classes of block codes. Here, by using the inclusion and exclusion principle, an explicit formula is derived which enumerates the complete weight distribution of an (n,k,d) linear code using a partially known weight distribution. This expression is analogous to the Pless power-moment identities - a system of equations relating the weight distribution of a linear code to the weight distribution of its dual code. Also, an approximate formula for the weight distribution of most linear (n,k,d) codes is derived. It is shown that for a given linear (n,k,d) code over GF(q), the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q = q(-)(n-k) as u becomes large. A relationship between the randomness of a linear block code and the minimum distance of its dual code is given, and it is shown that most linear block codes with rigid algebraic and combinatorial structure also display certain random properties which make them similar to random codes with no structure at all.

Cheung, K.-M.↗
NASA NTRS1992

Binary weight distributions of some Reed-Solomon codes

The binary weight distributions of the (7,5) and (15,9) Reed-Solomon (RS) codes and their duals are computed using the MacWilliams identities. Several mappings of symbols to bits are considered and those offering the largest binary minimum distance are found. These results are then used to compute bounds on the soft-decoding performance of these codes in the presence of additive Gaussian noise. These bounds are useful for finding large binary block codes with good performance and for verifying the performance obtained by specific soft-coding algorithms presently under development.

Pollara, F.↗
NASA NTRS1974

The subsequence weight distribution of summed maximum length digital sequences

An attempt is made to develop mathematical formulas to provide the basis for the design of pseudorandom signals intended for applications requiring accurate knowledge of the statistics of the signals. The analysis approach involves calculating the first five central moments of the weight distribution of subsequences of hybrid-sum sequences. The hybrid-sum sequence is formed from the modulo-two sum of k maximum length sequences and is an extension of the sum sequences formed from two maximum length sequences that Gilson (1966) evaluated. The weight distribution of the subsequences serves as an approximation to the filtering process. The basic reason for the analysis of hybrid-sum sequences is to establish a large group of sequences with good statistical properties. It is shown that this can be accomplished much more efficiently using the hybrid-sum approach rather than forming the group strictly from maximum length sequences.

Weathers, G. D.↗
NASA NTRS1989

Determinations of molecular weight and molecular weight distribution of high polymers by the rheological properties

Several methods are reviewed by which the molecular weight (MW) and the molecular weight distribution (MWD) of polymeric material were determined from the rheological properties. A poly(arylene ether) polymer with six different molecular weights was used in this investigation. Experimentally measured MW and MWD were conducted by GPC/LALLS (gel permeation chromatography/low angle laser light scattering), and the rheological properties of the melts were measured by a Rheometric System Four rheometer. It was found that qualitative information of the MW and MWD of these polymers could be derived from the viscoelastic properties, with the methods proposed by Zeichner and Patel, and by Dormier et al., by shifting the master curves of the dynamic storage modulus, G', and the loss modulus, G'', along the frequency axis. Efforts were also made to calculate quantitative profiles of MW and MWD for these polymers from their rheological properties. The technique recently proposed by Wu was evaluated. It was found that satisfactory results could only be obtained for polymers with single modal distribution in the molecular weight.

Huang, J. Y.↗
NASA NTRS1995

Weight distributions for turbo codes using random and nonrandom permutations

This article takes a preliminary look at the weight distributions achievable for turbo codes using random, nonrandom, and semirandom permutations. Due to the recursiveness of the encoders, it is important to distinguish between self-terminating and non-self-terminating input sequences. The non-self-terminating sequences have little effect on decoder performance, because they accumulate high encoded weight until they are artificially terminated at the end of the block. From probabilistic arguments based on selecting the permutations randomly, it is concluded that the self-terminating weight-2 data sequences are the most important consideration in the design of constituent codes; higher-weight self-terminating sequences have successively decreasing importance. Also, increasing the number of codes and, correspondingly, the number of permutations makes it more and more likely that the bad input sequences will be broken up by one or more of the permuters. It is possible to design nonrandom permutations that ensure that the minimum distance due to weight-2 input sequences grows roughly as the square root of (2N), where N is the block length. However, these nonrandom permutations amplify the bad effects of higher-weight inputs, and as a result they are inferior in performance to randomly selected permutations. But there are 'semirandom' permutations that perform nearly as well as the designed nonrandom permutations with respect to weight-2 input sequences and are not as susceptible to being foiled by higher-weight inputs.

Dolinar, S.↗
NASA NTRS1985

On the binary weight distribution of some Reed-Solomon codes

Consider an (n,k) linear code with symbols from GF(2 sup M). If each code symbol is represented by a m-tuple over GF(2) using certain basis for GF(2 sup M), a binary (nm,km) linear code is obtained. The weight distribution of a binary linear code obtained in this manner is investigated. Weight enumerators for binary linear codes obtained from Reed-Solomon codes over GF(2 sup M) generated by polynomials, (X-alpha), (X-l)(X-alpha), (X-alpha)(X-alpha squared) and (X-l)(X-alpha)(X-alpha squared) and their extended codes are presented, where alpha is a primitive element of GF(2 sup M). Binary codes derived from Reed-Solomon codes are often used for correcting multiple bursts of errors.

Lin, S.↗
NASA NTRS1987

The binary weight distribution of the extended (2 sup m, 2 sup m-4) code of Reed-Solomon code over GF(2 sup m) with generator polynomial (x-alpha sup 2) (x-alpha sup 3)

Consider an (n,k) linear code with symbols from GF(2 sup m). If each code symbol is represented by a binary m-tuple using a certain basis for GF(2 sup m), a binary (nm,km) linear code called a binary image of the original code is obtained. A lower bound is presented on the minimum weight enumerator for a binary image of the extended (2 sup m, 2 sup m -4) code of Reed-Solomon code over GF(2 sup m) with generator polynomical (x - alpha)(x- alpha squared)(x - alpha cubed) and its dual code, where alpha is a primitive element in GF(2 sup m).

Lin, Shu↗
NASA NTRS1946

Distribution of Structural Weight of Wing Along the Span

In the present report the true weight distribution law of the wing structure along the span is investigated. It is shown that the triangular distribution and that based on the proportionality to the chords do not correspond to the actual weight distribution, On the basis of extensive data on wings of the CAHI type airplane formulas are obtained from which it is possible to determine the true diagram of the structural weight distribution along the span from a knowledge of only the geometrical dimensions of the wing. At the end of the paper data are presented showing how the structural weight is distributed between the straight center portion and the tapered portion as a function of their areas.

Savelyev, V. V.↗
NASA NTRS1980

Vehicle systems design optimization study

The optimum vehicle configuration and component locations are determined for an electric drive vehicle based on using the basic structure of a current production subcompact vehicle. The optimization of an electric vehicle layout requires a weight distribution in the range of 53/47 to 62/38 in order to assure dynamic handling characteristics comparable to current internal combustion engine vehicles. Necessary modification of the base vehicle can be accomplished without major modification of the structure or running gear. As long as batteries are as heavy and require as much space as they currently do, they must be divided into two packages, one at front under the hood and a second at the rear under the cargo area, in order to achieve the desired weight distribution. The weight distribution criteria requires the placement of batteries at the front of the vehicle even when the central tunnel is used for the location of some batteries. The optimum layout has a front motor and front wheel drive. This configuration provides the optimum vehicle dynamic handling characteristics and the maximum passenger and cargo space for a given size vehicle.

Gilmour, J. L.↗
NASA NTRS1999

Investigation of Near Shannon Limit Coding Schemes

Turbo codes can deliver performance that is very close to the Shannon limit. This report investigates algorithms for convolutional turbo codes and block turbo codes. Both coding schemes can achieve performance near Shannon limit. The performance of the schemes is obtained using computer simulations. There are three sections in this report. First section is the introduction. The fundamental knowledge about coding, block coding and convolutional coding is discussed. In the second section, the basic concepts of convolutional turbo codes are introduced and the performance of turbo codes, especially high rate turbo codes, is provided from the simulation results. After introducing all the parameters that help turbo codes achieve such a good performance, it is concluded that output weight distribution should be the main consideration in designing turbo codes. Based on the output weight distribution, the performance bounds for turbo codes are given. Then, the relationships between the output weight distribution and the factors like generator polynomial, interleaver and puncturing pattern are examined. The criterion for the best selection of system components is provided. The puncturing pattern algorithm is discussed in detail. Different puncturing patterns are compared for each high rate. For most of the high rate codes, the puncturing pattern does not show any significant effect on the code performance if pseudo - random interleaver is used in the system. For some special rate codes with poor performance, an alternative puncturing algorithm is designed which restores their performance close to the Shannon limit. Finally, in section three, for iterative decoding of block codes, the method of building trellis for block codes, the structure of the iterative decoding system and the calculation of extrinsic values are discussed.

Kwatra, S. C.↗
NASA NTRS1980

Influence of excess diamine on properties of PMR polyimide resins and composites

By varying the stoichiometry of the reactants in the preparation of PMR polyimide resin, changes occur in molecular weight distribution which influence the rheological properties and thus the processability of the resin, as well as the mechanical properties of the composite. The influence of 1-10 percent molar excess MDA on the molecular weight distribution and rheological properties of an imidized PMR system were exposed. Molecular weight distribution is characterized by gel permeation chromatography of the imidized molding compound; shear viscosity is related to changes in average molecular weight. The thermo-oxidative stability at 600 F, glass transition temperature, flexural and interlaminar shear properties of PMR polyimide/Celion 6000 graphite fiber composites are compared as a function of the percent excess MDA in the monomer reactant mixture.

Hurwitz, F. I.↗
NASA NTRS1980

Influence of excess diamine on properties of PMR polyimide resins and composites

This preliminary study explores the influence of 1-10 percent molar excess MDA on the molecular weight distribution and rheological properties of an imidized PMR system. Molecular weight distribution is characterized by gel permeation chromatography of the imidized molding compound; shear viscosity is related to changes in average molecular weight. The thermo-oxidative stability at 600 F glass transition temperature, flexural and interlaminar shear properties of PMR polyimide/Celion 6000 graphite fiber composites are compared as a function of the percent excess MDA in the monomer reactant mixture.

Hurwitz, F. I.↗
NASA NTRS1987

Bandwidth efficient block codes for M-ary PSK modulation

A class of bandwidth efficient block codes for M-ary PSK modulation is presented. A soft-decision decoding for this class of codes is devised. Some specific short codes for Quad Phase Shift Key (QPSK), 8-PSK and 16-PSK modulations are constructed. These codes have good minimum squared Euclidean distances and provide 2 to 5.8 dB coding gains over uncoded QPSK modulation without (or with little) bandwidth expansion. The complete weight distributions of these specific codes are determined. Based on these weight distributions, their error probabilities are evaluated. Some of these codes have simple trellis structures and hence can be decoded by Viterbi decoding algorithm with relatively simple implementation. Moreover, the codes are very suitable for use as inner codes for various cascaded coding schemes with Reed-Solomon codes as outer codes.

Lin, Shu↗
NASA NTRS1992

On the decoder error probability of block codes

By using coding and combinational techniques, an explicit formula is derived which enumerates the complete weight distribution of decodable words of block codes using partially known weight distributions. Also an approximation formula for nonbinary block codes is obtained. These results in turn give exact and approximate expressions for the decoder error probability PE(u) of block codes.

Cheung, Kar-Ming↗
NASA NTRS1995

Summary of GPC/DV results for space exposed poly(arylene ether phosphine oxide)s

Gel Permeation Chromatography (GPC) was used to analyze poly(arylene ether phosphine oxide)s whose backbones were identical except for the ketone content and placement. These samples were exposed to low Earth orbit environment (predominantly atomic oxygen) on space shuttle flights. The materials and their unexposed controls were then characterized by GPC to investigate the effect of atomic oxygen on the molecular weight distributions. Analysis of the soluble portion of the samples revealed that there was significant loss of high molecular weight species. The presence of insoluble material also suggested that crosslinking was induced by the atomic oxygen exposure and that this very likely occurred at the high molecular weight portion of the molecular weight distribution.

Siochi, Emilie↗
NASA NTRS1940

The aileron as an aid to recovery from the spin

As part of a general investigation by the NACA of factors that affect the spin, the use of the aileron as an aid to recovery from the spin was studied. Tests of 10 different models, covering a wide range of mass distribution, were made in the NACA free-spinning tunnel to determine the effects of a large downward deflection of the outboard aileron and of normal angular deflections of the ailerons upon recovery characteristics. The results indicate that the direction of aileron setting, with or against the spin, which will aid recovery from the spin depends upon the airplane weight distribution. For monoplanes and for biplanes with lower-wing ailerons, ailerons with the spin will be favorable when the weight is distributed chiefly along the fuselage (single-engine airplanes) and ailerons against the spin will be favorable when the weight is distributed chiefly along the wings (multi engine airplanes). Downward movement of the outboard aileron through a large angle will not always be effective in aiding recovery, the effectiveness of such a movement also being dependent upon the weight distribution of the airplane.

Neihouse, A I↗