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All sources NASA NTRS DOE OSTI

NASA NTRS1990

New multi-level codes over GF(q)

Set partitioning to multi-dimensional signal spaces over GF(q), particularly GF sup q-1(q) and GF sup q (q), and show how to construct both multi-level block codes and multi-level trellis codes over GF(q). Two classes of multi-level (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum distance d sub min greater than or = d, are presented. These two classes of codes use Reed-Solomon codes as component codes. They can be easily decoded as block length q-1 Reed-Solomon codes or block length q or q + 1 extended Reed-Solomon codes using multi-stage decoding. Many of these codes have larger distances than comparable q-ary block codes, as component codes. Low rate q-ary convolutional codes, work error correcting convolutional codes, and binary-to-q-ary convolutional codes can also be used to construct multi-level trellis codes over GF(q) or binary-to-q-ary trellis codes, some of which have better performance than the above block codes. All of the new codes have simple decoding algorithms based on hard decision multi-stage decoding.

Wu, Jiantian↗

NASA NTRS1992

New multilevel codes over GF(q)

Wu, Jiantian↗

NASA NTRS1990

Performance comparison of combined ECC/RLL codes

In this paper, we present a performance comparison of several combined error correcting/run-lenth limited (ECC/RLL) codes created by concatenating a convolutional code with a run-length limited code. In each case, encoding and decoding are accomplished using a single trellis based on the combined code. Half of the codes under investigation use conventionally (d,k) run-length limited codes, where d is the minimum and k is the maximum allowable run of 0's between 1's. The other half of the combined codes use a special class of (d,k) codes known as distance preserving codes. These codes have the property that pairwise Hamming distances out of the (d,k) encoder are at least as large as the corresponding distances into the encoder (i.e., the codes preserve distance). Thus a combined code, created using a convolutional code concatenated with a distance preserving (d,k) code, will have a free distance (dfree) no smaller than the free distance of the original convolutional code. It should be noted that this does not hold if the (d,k) code was not distance preserving. A computer simulation is used to compare the performance of these two types of codes over the binary symmetric channel for various (d,k) constraints, rates, free distances, and numbers of states. Of particular interest for magnetic recording applications are codes with run-length constraints (1,3), (1,7), and (2,7).

French, C.↗

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 NTRS2004

Accumulate repeat accumulate codes

In this paper we propose an innovative channel coding scheme called 'Accumulate Repeat Accumulate codes' (ARA). This class of codes can be viewed as serial turbo-like codes, or as a subclass of Low Density Parity Check (LDPC) codes, thus belief propagation can be used for iterative decoding of ARA codes on a graph. The structure of encoder for this class can be viewed as precoded Repeat Accumulate (RA) code or as precoded Irregular Repeat Accumulate (IRA) code, where simply an accumulator is chosen as a precoder. Thus ARA codes have simple, and very fast encoder structure when they representing LDPC codes. Based on density evolution for LDPC codes through some examples for ARA codes, we show that for maximum variable node degree 5 a minimum bit SNR as low as 0.08 dB from channel capacity for rate 1/2 can be achieved as the block size goes to infinity. Thus based on fixed low maximum variable node degree, its threshold outperforms not only the RA and IRA codes but also the best known LDPC codes with the dame maximum node degree. Furthermore by puncturing the accumulators any desired high rate codes close to code rate 1 can be obtained with thresholds that stay close to the channel capacity thresholds uniformly. Iterative decoding simulation results are provided. The ARA codes also have projected graph or protograph representation that allows for high speed decoder implementation.

Low Density Parity Check codes (LDPC)↗

DOE OSTI2020

Local Truncation Error-Informed Code Verification

The method of manufactured solutions (MMS) has become increasingly popular in conducting code verification studies on predictive codes, such as nuclear power system codes and computational fluid dynamic codes. The reason for the popularity of this approach is that it can be used when an analytical solution is not available. Using MMS, code developers are able to verify that their code is free of coding errors that impact the observed order of accuracy. While MMS is still an excellent tool for code verification, it does not identify coding errors that are of the same order as the numerical method. This paper presents a method that combines MMS with modified equation analysis (MEA), which calculates the local truncation error (LTE) to identify coding error up to and including the order of the numerical method. This method is referred to as modified equation analysis methd of manufactured solutions (MEAMMS). MEAMMS is then applied to a custom-built code, which solves the shallow water equations, to test the performance of the code verification method. MEAMMS is able to detect all coding errors that impact the implementation of the numerical scheme. To show how MEAMMS is different than MMS, they are both applied to the same first-order numerical method test problem with a first-order coding error. When there are first-order coding errors, only MEAMMS is able to identify them. Finally, this shows that MEAMMS is able to identify a larger set of coding errors while still being able to identify the coding errors MMS is able to identify.

42 ENGINEERING↗

NASA NTRS2010

Performance Bounds on Two Concatenated, Interleaved Codes

A method has been developed of computing bounds on the performance of a code comprised of two linear binary codes generated by two encoders serially concatenated through an interleaver. Originally intended for use in evaluating the performances of some codes proposed for deep-space communication links, the method can also be used in evaluating the performances of short-block-length codes in other applications. The method applies, more specifically, to a communication system in which following processes take place: At the transmitter, the original binary information that one seeks to transmit is first processed by an encoder into an outer code (Co) characterized by, among other things, a pair of numbers (n,k), where n (n > k)is the total number of code bits associated with k information bits and n k bits are used for correcting or at least detecting errors. Next, the outer code is processed through either a block or a convolutional interleaver. In the block interleaver, the words of the outer code are processed in blocks of I words. In the convolutional interleaver, the interleaving operation is performed bit-wise in N rows with delays that are multiples of B bits. The output of the interleaver is processed through a second encoder to obtain an inner code (Ci) characterized by (ni,ki). The output of the inner code is transmitted over an additive-white-Gaussian- noise channel characterized by a symbol signal-to-noise ratio (SNR) Es/No and a bit SNR Eb/No. At the receiver, an inner decoder generates estimates of bits. Depending on whether a block or a convolutional interleaver is used at the transmitter, the sequence of estimated bits is processed through a block or a convolutional de-interleaver, respectively, to obtain estimates of code words. Then the estimates of the code words are processed through an outer decoder, which generates estimates of the original information along with flags indicating which estimates are presumed to be correct and which are found to be erroneous. From the perspective of the present method, the topic of major interest is the performance of the communication system as quantified in the word-error rate and the undetected-error rate as functions of the SNRs and the total latency of the interleaver and inner code. The method is embodied in equations that describe bounds on these functions. Throughout the derivation of the equations that embody the method, it is assumed that the decoder for the outer code corrects any error pattern of t or fewer errors, detects any error pattern of s or fewer errors, may detect some error patterns of more than s errors, and does not correct any patterns of more than t errors. Because a mathematically complete description of the equations that embody the method and of the derivation of the equations would greatly exceed the space available for this article, it must suffice to summarize by reporting that the derivation includes consideration of several complex issues, including relationships between latency and memory requirements for block and convolutional codes, burst error statistics, enumeration of error-event intersections, and effects of different interleaving depths. In a demonstration, the method was used to calculate bounds on the performances of several communication systems, each based on serial concatenation of a (63,56) expurgated Hamming code with a convolutional inner code through a convolutional interleaver. The bounds calculated by use of the method were compared with results of numerical simulations of performances of the systems to show the regions where the bounds are tight (see figure).

Moision, Bruce↗

DOE OSTI2018

Final Energy Savings Analysis of the Proposed NYStretch-Energy Code 2018

This study was conducted by the Pacific Northwest National Laboratory (PNNL) in support of the stretch energy code development led by the New York State Energy Research and Development Authority (NYSERDA). In 2017 NYSERDA developed its 2016 Stretch Code Supplement to the 2016 New York State Energy Conservation Construction Code (hereinafter referred to as “NYStretch-Energy”). NYStretch-Energy is intended as a model energy code for statewide voluntary adoption that anticipates other code advancements culminating in the goal of a statewide Net Zero Energy Code by 2028. Since then, NYSERDA continues to develop the NYStretch-Energy Code 2018 edition (NYSERDA 2018). To support the effort, PNNL conducted energy simulation analysis to quantify the energy savings of proposed commercial provisions of the NYStretch-Energy Code 2018 in New York. The focus of this project is to significantly improve the energy efficiency over existing commercial model energy codes. A key requirement of the proposed stretch code is that it be ‘adoptable’ as an energy code, meaning that it must align with current code scope and limitations, and primarily impact building components that are currently regulated by local building departments. It is largely limited to prescriptive measures, which are what most building departments and design projects are most familiar with. This report describes a set of energy-efficiency measures (EEMs) that demonstrate significant energy savings over ANSI/ASHRAE/IES Standard 90.1-2013 (ASHRAE 2013) across a broad range of commercial building types and all three climate zones in New York. In collaboration with New Building Institute, the EEMs were developed from national model codes and standards, high-performance building codes and standards, regional energy codes, and measures being proposed as part of the on-going code development process. PNNL analyzed these measures using whole building energy models for selected prototype commercial buildings and multifamily buildings representing buildings in New York.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

NASA NTRS1994

Trace-shortened Reed-Solomon codes

Reed-Solomon (RS) codes have been part of standard NASA telecommunications systems for many years. RS codes are character-oriented error-correcting codes, and their principal use in space applications has been as outer codes in concatenated coding systems. However, for a given character size, say m bits, RS codes are limited to a length of, at most, 2(exp m). It is known in theory that longer character-oriented codes would be superior to RS codes in concatenation applications, but until recently no practical class of 'long' character-oriented codes had been discovered. In 1992, however, Solomon discovered an extensive class of such codes, which are now called trace-shortened Reed-Solomon (TSRS) codes. In this article, we will continue the study of TSRS codes. Our main result is a formula for the dimension of any TSRS code, as a function of its error-correcting power. Using this formula, we will give several examples of TSRS codes, some of which look very promising as candidate outer codes in high-performance coded telecommunications systems.

Mceliece, R. J.↗

NASA NTRS1999

An Interactive Concatenated Turbo Coding System

This paper presents a concatenated turbo coding system in which a Reed-Solomon outer code is concatenated with a binary turbo inner code. In the proposed system, the outer code decoder and the inner turbo code decoder interact to achieve both good bit error and frame error performances. The outer code decoder helps the inner turbo code decoder to terminate its decoding iteration while the inner turbo code decoder provides soft-output information to the outer code decoder to carry out a reliability-based soft- decision decoding. In the case that the outer code decoding fails, the outer code decoder instructs the inner code decoder to continue its decoding iterations until the outer code decoding is successful or a preset maximum number of decoding iterations is reached. This interaction between outer and inner code decoders reduces decoding delay. Also presented in the paper are an effective criterion for stopping the iteration process of the inner code decoder and a new reliability-based decoding algorithm for nonbinary codes.

Liu, Ye↗

NASA NTRS2000

Error Control Coding Techniques for Space and Satellite Communications

This paper presents a concatenated turbo coding system in which a Reed-Solomom outer code is concatenated with a binary turbo inner code. In the proposed system, the outer code decoder and the inner turbo code decoder interact to achieve both good bit error and frame error performances. The outer code decoder helps the inner turbo code decoder to terminate its decoding iteration while the inner turbo code decoder provides soft-output information to the outer code decoder to carry out a reliability-based soft-decision decoding. In the case that the outer code decoding fails, the outer code decoder instructs the inner code decoder to continue its decoding iterations until the outer code decoding is successful or a preset maximum number of decoding iterations is reached. This interaction between outer and inner code decoders reduces decoding delay. Also presented in the paper are an effective criterion for stopping the iteration process of the inner code decoder and a new reliability-based decoding algorithm for nonbinary codes.

Lin, Shu↗

DOE OSTI2015

SURVEY OF COUPLING SCHEMES IN TRADITIONAL COUPLED NEUTRONICS AND THERMAL-HYDRAULICS CODES

The design and the analysis of nuclear power plants (NPPs) require computational codes to predict the behavior of the NPP nuclear components and other systems (i.e., reactor core, primary coolant system, emergency core cooling system, etc.). Coupled calculations are essential to the conduct of deterministic safety assessments. Inasmuch as the physical phenomena that govern the performance of a nuclear reactor are always present simultaneously, ideally computational modeling of a nuclear reactor should include coupled codes that represent all of the active physical phenomena. Such multi-physics codes are under development at several institutions and are expected to become operational in the future. However, in the interim, integrated codes that incorporate modeling capabilities for two to three physical phenomena will remain useful. For example, in the conduct of safety analyses, of paramount importance are codes that couple neutronics and thermal-hydraulics, especially transient codes. Other code systems of importance to safety analyses are those that couple primary system thermal-hydraulics to fission product chemistry, neutronics to fuel performance, containment behavior and structural mechanics to thermal-hydraulics, etc. This paper surveys the methods used traditionally in the coupling of neutronic and thermal-hydraulics codes. The neutron kinetics codes are used for computing the space-time evolution of the neutron flux and, hence, of the power distribution. The thermal-hydraulics codes, which compute mass, momentum and energy transfers, model the coolant flow and the temperature distribution. These codes can be used to compute the neutronic behavior and the thermal-hydraulic states separately. However, the need to account with fidelity for the dynamic feedback between the two sets of properties (via temperature and density effects on the cross section inputs into the neutronics codes) and the requirement to model realistically the transient response of nuclear power plants and to assess the associated emergency systems and procedures imply the necessity of modeling the neutronic and thermal-hydraulics simultaneously within a coupled code system. The focus of this paper is a comparison of the methods by which the coupling between neutron kinetics and thermal-hydraulics treatments has been traditionally achieved in various code systems. As discussed in the last section, the modern approaches to multi-physics code development are beyond the scope of this paper.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2011

Renewable Energy Requirements for Future Building Codes: Options for Compliance

As the model energy codes are improved to reach efficiency levels 50 percent greater than current codes, use of on-site renewable energy generation is likely to become a code requirement. This requirement will be needed because traditional mechanisms for code improvement, including envelope, mechanical and lighting, have been pressed to the end of reasonable limits. Research has been conducted to determine the mechanism for implementing this requirement (Kaufman 2011). Kaufmann et al. determined that the most appropriate way to structure an on-site renewable requirement for commercial buildings is to define the requirement in terms of an installed power density per unit of roof area. This provides a mechanism that is suitable for the installation of photovoltaic (PV) systems on future buildings to offset electricity and reduce the total building energy load. Kaufmann et al. suggested that an appropriate maximum for the requirement in the commercial sector would be 4 W/ft 2 of roof area or 0.5 W/ft 2 of conditioned floor area. As with all code requirements, there must be an alternative compliance path for buildings that may not reasonably meet the renewables requirement. This might include conditions like shading (which makes rooftop PV arrays less effective), unusual architecture, undesirable roof pitch, unsuitable building orientation, or other issues. In the short term, alternative compliance paths including high performance mechanical equipment, dramatic envelope changes, or controls changes may be feasible. These options may be less expensive than many renewable systems, which will require careful balance of energy measures when setting the code requirement levels. As the stringency of the code continues to increase however, efficiency trade-offs will be maximized, requiring alternative compliance options to be focused solely on renewable electricity trade-offs or equivalent programs. One alternate compliance path includes purchase of Renewable Energy Credits (RECs). Each REC represents a specified amount of renewable electricity production and provides an offset of environmental externalities associated with non-renewable electricity production. The purpose of this paper is to explore the possible issues with RECs and comparable alternative compliance options. Existing codes have been examined to determine energy equivalence between the energy generation requirement and the RECs alternative over the life of the building. The price equivalence of the requirement and the alternative are determined to consider the economic drivers for a market decision. This research includes case studies that review how the few existing codes have incorporated RECs and some of the issues inherent with REC markets. Section 1 of the report reviews compliance options including RECs, green energy purchase programs, shared solar agreements and leases, and other options. Section 2 provides detailed case studies on codes that include RECs and community based alternative compliance methods. The methods the existing code requirements structure alternative compliance options like RECs are the focus of the case studies. Section 3 explores the possible structure of the renewable energy generation requirement in the context of energy and price equivalence. The price of RECs have shown high variation by market and over time which makes it critical to for code language to be updated frequently for a renewable energy generation requirement or the requirement will not remain price-equivalent over time. Section 4 of the report provides a maximum case estimate for impact to the PV market and the REC market based on the Kaufmann et al. proposed requirement levels. If all new buildings in the commercial sector complied with the requirement to install rooftop PV arrays, nearly 4,700 MW of solar would be installed in 2012, a major increase from EIA estimates of 640 MW of solar generation capacity installed in 2009. The residential sector could contribute roughly an additional 2,300 MW based on the same code requirement levels of 4 W/ft 2 of roof area. Section 5 of the report provides a basic framework for draft code language recommendations based on the analysis of the alternative compliance levels.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2013

Four-Dimensional Finite-Orbit-Width Fokker-Planck Code with Sources, for Neoclassical/Anomalous Transport Simulation of Ion and Electron (Final Technical Report)

Within the US Department of Energy/Office of Fusion Energy magnetic fusion research program, there is an important whole-plasma-modeling need for a radio-frequency/neutral-beam-injection (RF/NBI) transport-oriented finite-difference Fokker-Planck (FP) code with combined capabilities for 4D (2R2V) geometry near the fusion plasma periphery, and computationally less demanding 3D (1R2V) bounce-averaged capabilities for plasma in the core of fusion devices. Demonstration of proof-of-principle achievement of this goal has been carried out in research carried out under Phase I of the SBIR award. Two DOE-sponsored codes, the CQL3D bounce-average Fokker-Planck code in which CompX has specialized, and the COGENT 4D, plasma edge-oriented Fokker-Planck code which has been constructed by Lawrence Livermore National Laboratory and Lawrence Berkeley Laboratory scientists, where coupled. Coupling was achieved by using CQL3D calculated velocity distributions including an energetic tail resulting from NBI, as boundary conditions for the COGENT code over the two-dimensional velocity space on a spatial interface (flux) surface at a given radius near the plasma periphery. The finite-orbit-width fast ions from the CQL3D distributions penetrated into the peripheral plasma modeled by the COGENT code. This combined code demonstrates the feasibility of the proposed 3D/4D code. By combining these codes, the greatest computational efficiency is achieved subject to present modeling needs in toroidally symmetric magnetic fusion devices. The more efficient 3D code can be used in its regions of applicability, coupled to the more computationally demanding 4D code in higher collisionality edge plasma regions where that extended capability is necessary for accurate representation of the plasma. More efficient code leads to greater use and utility of the model. An ancillary aim of the project is to make the combined 3D/4D code user friendly. Achievement of full-coupling of these two Fokker-Planck codes will advance computational modeling of plasma devices important to the USDOE magnetic fusion energy program, in particular the DIII-D tokamak at General Atomics, San Diego, the NSTX spherical tokamak at Princeton, New Jersey, and the MST reversed-field-pinch Madison, Wisconsin. The validation studies of the code against the experiments will improve understanding of physics important for magnetic fusion, and will increase our design capabilities for achieving the goals of the International Tokamak Experimental Reactor (ITER) project in which the US is a participant and which seeks to demonstrate at least a factor of five in fusion power production divided by input power.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2004

Maximum likelihood decoding analysis of Accumulate-Repeat-Accumulate Codes

Repeat-Accumulate (RA) codes are the simplest turbo-like codes that achieve good performance. However, they cannot compete with Turbo codes or low-density parity check codes (LDPC) as far as performance is concerned. The Accumulate Repeat Accumulate (ARA) codes, as a subclass of LDPC codes, are obtained by adding a pre-coder in front of RA codes with puncturing where an accumulator is chosen as a precoder. These codes not only are very simple, but also achieve excellent performance with iterative decoding. In this paper, the performance of these codes with (ML) decoding are analyzed and compared to random codes by very tight bounds. The weight distribution of some simple ARA codes is obtained, and through existing tightest bounds we have shown the ML SNR threshold of ARA codes approaches very closely to the performance of random codes. We have shown that the use of precoder improves the SNR threshold but interleaving gain remains unchanged with respect to RA code with puncturing.

low density parity codes (LDPC)↗

DOE OSTI2025

Fault Tolerant Decoding of QLDPC-GKP Codes with Circuit Level Soft Information

Concatenated bosonic-stabilizer codes have recently gained prominence as promising candidates for achieving low-overhead fault-tolerant quantum computing in the long term. In such systems, analog information obtained from the syndrome measurements of an inner bosonic code is used to inform decoding for an outer code layer consisting of a discrete-variable stabilizer code such as a surface code. The use of Quantum Low-Density Parity Check (QLDPC) codes as an outer code is of particular interest due to the significantly higher encoding rates offered by these code families, leading to a further reduction in overhead for large-scale quantum computing. Recent works have investigated the performance of QLDPC-GKP codes in detail, and the use of analog information from the inner code significantly boosts decoder performance. However, the noise models assumed in these works are typically limited to depolarizing or phenomenological noise. In this paper, we investigate the performance of QLDPC-GKP concatenated codes under circuit-level noise, based on a model introduced by Noh et al. in the context of the surface-GKP code. To demonstrate the performance boost from analog information, we investigate three scenarios: (a) decoding without soft information, (b) decoding with precomputed error probabilities but without real-time soft information, and (c) decoding with real-time soft information obtained from round-to-round decoding of the inner GKP code. Results show minimal improvement between (a) and (b), but a significant boost in (c), indicating that real-time soft information is critical for concatenated decoding under circuit-level noise. We also study the effect of measurement schedules with varying depths and show that using a schedule with minimum depth is essential for obtaining reliable soft information from the inner code.

Borah, Shantom K. [Arizona U. (main)]↗

NASA NTRS1988

A new code for Galileo

Over the past six to eight years, an extensive research effort was conducted to investigate advanced coding techniques which promised to yield more coding gain than is available with current NASA standard codes. The delay in Galileo's launch due to the temporary suspension of the shuttle program provided the Galileo project with an opportunity to evaluate the possibility of including some version of the advanced codes as a mission enhancement option. A study was initiated last summer to determine if substantial coding gain was feasible for Galileo and, is so, to recommend a suitable experimental code for use as a switchable alternative to the current NASA-standard code. The Galileo experimental code study resulted in the selection of a code with constant length 15 and rate 1/4. The code parameters were chosen to optimize performance within cost and risk constraints consistent with retrofitting the new code into the existing Galileo system design and launch schedule. The particular code was recommended after a very limited search among good codes with the chosen parameters. It will theoretically yield about 1.5 dB enhancement under idealizing assumptions relative to the current NASA-standard code at Galileo's desired bit error rates. This ideal predicted gain includes enough cushion to meet the project's target of at least 1 dB enhancement under real, non-ideal conditions.

Dolinar, S.↗

NASA NTRS1990

On linear structure and phase rotation invariant properties of block 2(sup l)-PSK modulation codes

Two important structural properties of block 2(l)-ary PSK (phase shift keying) modulation codes, linear structure and phase symmetry, are investigated. For an additive white Gaussian noise (AWGN) channel, the error performance of a modulation code depends on its squared Euclidean distance distribution. Linear structure of a code makes the error performance analysis much easier. Phase symmetry of a code is important in resolving carrier phase ambiguity and ensuring rapid carrier phase resynchronization after temporary loss of synchronization. It is desirable for a code to have as many phase symmetries as possible. A 2(l)-ary modulation code is represented here as a code with symbols from the integer group. S sub 2(l) PSK = (0,1,2,...,2(l)-1), under the modulo-2(l) addition. The linear structure of block 2(l)-ary PSK modulation codes over S sub 2(l)-ary PSK with respect to the modulo-2(l) vector addition is defined, and conditions under which a block 2(l)-ary PSK modulation code is linear are derived. Once the linear structure is developed, phase symmetry of a block 2(l)-ary PSK modulation code is studied. It is a necessary and sufficient condition for a block 2(l)-PSK modulation code, which is linear as a binary code, to be invariant under 180 deg/2(l-h) phase rotation, for 1 is less than or equal to h is less than or equal to l. A list of short 8-PSK and 16-PSK modulation codes is given, together with their linear structure and the smallest phase rotation for which a code is invariant.

Lin, Shu↗