On multilevel block modulation codes

The multilevel technique for combining block coding and modulation is investigated. A general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures. A specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed. Given a multilevel block modulation code C with no interdependency among the binary component codes, the proposed method gives a multilevel block modulation code C' that has the same rate as C, a minimum squared Euclidean distance not less than that of C, a trellis diagram with the same number of states as that of C, and a smaller number of nearest neighbor codewords than that of C . Finally, a technique is presented for analyzing the error performance of block modulation codes for an additive white Gaussian noise (AWGN) channel based on soft-decision maximum likelihood decoding. Error probabilities of some specific codes are evaluated by simulation and upper bounds based on their Euclidean weight distributions     

Bandwidth efficient coding for fading channels: code constructionand performance analysis

The authors apply a general method of bounding the event error probability of TCM (trellis-coded modulation) schemes to fading channels and use the effective length and the minimum-squared-product distance to replace the minimum-free-squared-Euclidean distance as code design parameters for Rayleigh and Rician fading channels with a substantial multipath component. They present 8-PSK (phase-shift-keying) trellis codes specifically constructed for fading channels that outperform equivalent codes designed for the AWGN (additive white Gaussian noise) channel when v⩾5. For quasiregular trellis codes there exists an efficient algorithm for evaluating event error probability, and numerical results which demonstrate the importance of the effective length as a code design parameter for fading channels with or without side information have been obtained. This is consistent with the case for binary signaling, where the Hamming distance remains the best code design parameter for fading channels. The authors show that the use of Reed-Solomon block codes with expanded signal sets becomes interesting only for large value of E_s/N₀, where they begin to outperform trellis codes     

Two 16-state, rate R=2/4 trellis codes whose free distances meetthe Heller bound

For rate R=1/2 convolutional codes with 16 states there exists a gap between Heller's (1968) upper bound on the free distance and its optimal value. This article reports on the construction of 16-state, binary, rate R=2/4 nonlinear trellis and convolutional codes having d _free=8; a free distance that meets the Heller upper bound. The nonlinear trellis code is constructed from a 16-state, rate R=1/2 convolutional code over Z₄ using the Gray map to obtain a binary code. Both convolutional codes are obtained by computer search. Systematic feedback encoders for both codes are potential candidates for use in combination with iterative decoding. Regarded as modulation codes for 4-PSK, these codes have free squared Euclidean distance d_{E, free}²=16     

Upper bounds for small trellis codes

An upper bound on the minimum squared distance of trellis codes by packing Voronoi cells is derived and compared with previously known bounds. The authors focus on codes with small memory for modulation formats such as pulse amplitude modulation (PAM), m-ary quadrature amplitude modulation (QAM), and m-ary phase shift keying (PSK). The bound is tight to search results for coset codes with a small number of states     

Improved coding techniques for preceded partial-response channels

A coset of a convolutional code may be used to generate a zero-run length limited trellis code for a 1-D partial-response channel. The free squared Euclidean distance, d_free², at the channel output is lower bounded by the free Hamming distance of the convolutional code. The lower bound suggests the use of a convolutional code with maximal free Hamming distance, d_max(R,N), for given rate R and number of decoder states N. In this paper we present cosets of convolutional codes that generate trellis codes with d_free ²>d_max(R,N) for rates 1/5⩽R⩽7/9 and (d _free²=d_max(R,N) for R=13/16,29/32,61/64, The tabulated convolutional codes with R⩽7/9 were not optimized for Hamming distance. Instead, a computer search was used to determine cosets of convolutional codes that exploit the memory of the 1-D channel to increase d_free² at the channel output. The search was limited by only considering cosets with certain structural properties. The R⩾13/16 codes were obtained using a new construction technique for convolutional codes with free Hamming distance 4. Newly developed bounds on the maximum zero-run lengths of cosets were used to ensure a short maximum run length at the 1-D channel output     

Rate-compatible punctured convolutional codes (RCPC codes) andtheir applications

The concept of punctured convolutional codes is extended by punctuating a low-rate 1/N code periodically with period P to obtain a family of codes with rate P/(P+l), where l can be varied between 1 and (N-1)P. A rate-compatibility restriction on the puncturing tables ensures that all code bits of high rate codes are used by the lower-rate codes. This allows transmission of incremental redundancy in ARQ/FEC (automatic repeat request/forward error correction) schemes and continuous rate variation to change from low to high error protection within a data frame. Families of RCPC codes with rates between 8/9 and 1/4 are given for memories M from 3 to 6 (8 to 64 trellis states) together with the relevant distance spectra. These codes are almost as good as the best known general convolutional codes of the respective rates. It is shown that the same Viterbi decoder can be used for all RCPC codes of the same M. the application of RCPC codes to hybrid ARQ/FEC schemes is discussed for Gaussian and Rayleigh fading channels using channel-state information to optimise throughput     

Trellis coding with multidimensional QAM signal sets

Trellis coding using multidimensional quadrature amplitude modulation (QAM) signal sets is investigated. Finite-size 2D signal sets are presented that have minimum average energy, are 90° rotationally symmetric, and have from 16 to 1024 points. The best trellis codes using the finite 16-QAM signal set with two, four, six, and eight dimensions are found by computer search (the multidimensional (multi-D) signal set is constructed from the 2-D signal set). The best moderate complexity trellis codes for infinite lattices with two, four six, and eight dimensions are also found. The minimum free squared Euclidean distance and number of nearest neighbors for these codes were used as the selection criteria. Many of the multi-D codes are fully rotationally invariant and give asymptotic coding gains up to 6.0 dB. From the infinite lattice codes, the best codes for transmitting J, J+1/4, J+1/3, J+1/2, J+2/3, and J+3/4 b/sym (J an integer) are presented     

New multilevel codes over GF(q)

Set partitioning is applied to multidimensional signal spaces over GF(q), i.e., GFⁿ¹(q) (n1⩽q ), and it is shown how to construct both multilevel block codes and multilevel trellis codes over GF(q). Multilevel (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum distance d_min⩾d are presented. These codes use Reed-Solomon codes as component codes. Longer multilevel block codes are also constructed using q-ary block codes with block length longer than q+1 as component codes. Some quaternary multilevel block codes are presented with the same length and number of information symbols as, but larger distance than, the best previously known quaternary one-level block codes. It is proved that if all the component block codes are linear. the multilevel block code is also linear. Low-rate q-ary convolutional codes, word-error-correcting convolutional codes, and binary-to-q-ary convolutional codes can also be used to construct multilevel trellis codes over GF(q) or binary-to-q-ary trellis codes     

On the influence of coding on the mean time to failure fordegrading memories with defects

The application of a combined test-error-correcting procedure is studied to improve the mean time to failure (MTTF) for degrading memory systems with defects. The degradation is characterized by the probability p that within a unit of time a memory cell changes from the operational state to the permanent defect state. Bounds are given on the MTTF and it is shown that, for memories with N words of k information bits, coding gives an improvement in MTTF proportional to (k/n) N(d_min^-2)/(d_min ^-1), where d_min and (k/n) are the minimum distance and the efficiency of the code used, respectively. Thus the time gain for a simple minimum-distance-3 is proportional to N^-1. A memory word test is combined with a simple defect-matching code. This yields reliable operation with one defect in a word of length k+2 at a code efficiency k/(k+2)     

New ternary line codes based on trellis structure

The trellis coding technique is applied to line-coded baseband digital transmission systems. For R=n/n+1(n=1,2,3) coding rates, a new codeword assignment model is proposed to accomplish basic requirements for line coding in which each length n binary data sequence is encoded into a length n+1 ternary (+,0,-) line codeword chosen among the code alphabet with 2ⁿ⁺² elements. Assuming Viterbi decoding, the system error performance is improved by increasing the free Euclidean distance between coded sequences. A new algorithm is given for the calculation of the free distance between line-coded sequences so obtained. For R=1/2 and R=3/4 rates, the analytical error performance upper bounds are derived. The power spectral densities of the new line codes are also calculated and compared with those of known line codes     

Channel coding with quadrature-quadrature phase shift-keying(Q2PSK) signals

Channel coding due to trellis modulation has been proved to be useful for bandlimited channels. However, these modulations, mostly designed with n-use of 2D signals, are primarily aimed at coding gain only. It is pointed out that utilization of all available signal dimensions, which is limited by the time-bandwidth product, may improve the bandwidth efficiency and simultaneously bring an additional coding gain. Trellis coding with a spectrally efficient 4D signal set based on Q²PSK is addressed. Without any expansion of the Q ²PSK signal set, a simple hand-designed 16-state trellis code provides a coding gain of 6.02 dB. With the same number of states in G. Ungergoeck's (1982) 8-PSK trellis, the gain is 4.1 dB. In low intersymbol interference situations, the bandwidth efficiency of this coded Q²PSK is twice that of coded 8-PSK; if both operate at 2 b/s Hz and P_b(E)=10^-5, the coded Q²PSK provides a saving of about 4 dB over the coded 8-PSK. Some fully connected trellises with an expanded signal due to 2-use of Q²PSK signals are also presented. Two such codes ar rate-7/8 achieve a gain of 5.45 dB with only eight states     

Achieving the designed error capacity in decodingalgebraic-geometric codes

A decoding algorithm for codes arising from algebraic curves explicitly constructable by Goppa's construction is presented. Any configuration up to the greatest integer less than or equal to (d *-1)/2 errors is corrected by the algorithm whenever d*⩾6g, where d* is the designed minimum distance of the code and g is the genus of the curve. The algorithm's complexity is at most O((d*)² n), where n denotes the length of the code. Application to Hermitian codes and connections with well-known algorithms are explained     

A semi-empirical model for the field-effect mobility ofhydrogenated polycrystalline-silicon MOSFETs

The quantitative relationship between field-effect mobility (μ _FE) and grain-boundary trap-state density (N_t ) in hydrogenated polycrystalline-silicon (poly-Si) MOSFETs is investigated. The focus is on the field-effect mobility in MOSFETs with N_t 1×10² cm^-2. It is found that reducing N_t to as low as 5×10¹¹ cm^-2 has a great impact on μ_FE. MOSFETs with the N_t of 4.2×10¹¹ cm^-2 show an electron mobility of 185 cm²/V-s, despite a mean grain size of 0.5 μm. The three principal factors that determine μ_FE, namely, the low-field mobility, the mobility degradation factor, and the trap-state density N_t are clarified     

Bulk and surface properties of amorphous hydrogenated fluorinatedsilicon grown from SiF4 and H2

A detailed study of the growth of amorphous hydrogenated fluorinated silicon (a-Si:H, F) from a DC glow discharge in SiF₄ and H₂ is discussed. The electrical properties of the films can be varied over a very wide range. The bulk properties of the best films that were measured included an Urbach energy E_u =43 meV, a deep-level defect density N_s=1.5×10¹⁵ cm^-3, and a hole drift mobility of 8×10^-3 cm² V^-1 s^-1, which reflects a characteristic valence band energy of 36 meV. It was found that E_u, N _s, and the density of surface states N_ss are related to each other. Under the deposition condition of the films with the best bulk properties, N_ss reaches its highest value of 1×10¹⁴ cm^-2. It is suggested that in growth from SiF₄/H₂, the density of dangling bonds at the growing surface is very sensitive to the deposition conditions     

The sublinear relationship between index change and carrier densityin 1.5 and 1.3 μm semiconductor lasers

The carrier-induced index change was measured using a novel injection-reflection technique in combination with differential carrier lifetime data. The observed relation between index change and injected carrier density at bandgap wavelength is nonlinear and is approximately given by δn_act=-6.1×10^-14 ( N)^0.66 for a 1.5-μm laser and δn _act=-1.3×10^-14 (N)^0.68 for a 1.3-μm laser. The carrier-induced index change for a 1.3-μm laser at 1.53-μm wavelength is smaller and is given by δn _act=-9.2×10^-16 (N)^0.72      

On the Hamming distance properties of group codes

Under certain mild conditions, the minimum Hamming distance D of an (N, K, D) group code C over a non-abelian group G is bounded by D⩽N -2K+2 if K⩽N/2, and is equal to 1 if K>N/2. Consequently, there exists no (N, K, N-K+1) group code C over an non-abelian group G if 1<K<N. Moreover, any normal code C with a non-abelian output space has minimum Hamming distance equal to D=1. These results follow from the fact that non-abelian groups have nontrivial commutator subgroups. Finally, if C is an (N, K, D) group code over an abelian group G that is not elementary abelian, then there exists an (N, K, D) group code over a smaller elementary abelian group G'. Thus, a group code over a general group G cannot have better parameters than a conventional linear code over a field of the same size as G     

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme is presented for error control in data communications. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary phase-shift keying (PSK) modulation. Error performance of the scheme is analyzed for an additive white Gaussian noise (AWGN) channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner-code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The scheme is particularly effective for high-speed satellite communications for large file transfer where high reliability is required. A simple method is also presented for constructing block codes for M-ary PSK modulation. Soome short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft-decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45° rotation     

The design of trellis coded MPSK for fading channels: performancecriteria

It has been well established that the appropriate criterion for optimum trellis-coded modulation design on the additive white Gaussian noise channel is maximization of the free Euclidean distance. It is shown that when the trellis-coded modulation is used on a Rician fading channel with interleaving/deinterleaving, the design of the code of optimum performance is guided by other factors, in particular, the length of the shortest error-event path, and the product of branch distances (possibly normalized by the Euclidean distance of the path) along the path. Although maximum free distance (d_free) is still an important consideration, it plays a less significant role the more severe the fading is on the channel. These considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over Rician fading channels. If no interleaving/deinterleaving is used, then once again the design of the trellis code is guided by maximizing d_free      

衰落信道下的格形码设计

衰落信道下ＴＣＭ好码的设计准则是使有效码长度最长，同时使其对应路径的欧几里德距离乘积最大。本文首先从理论上得到有效码长度与卷积编码器的状态数，并行输入数之间的关系。提出了一种能达到最大自由长的状态转移图－标准拓扑篱笆图的概念，在此基础上，对衰落信道下采用速率为（２／３）８ＰＳＫ信号集合时的ＴＣＭ好码进行搜索，与文献中已有码相比，利用准则判别和进行蒙特－卡洛模拟都说明了新码在抗衰落方面的良好性能。     

An Upper Bound on the Minimum Distance of Serially Concatenated Convolutional Codes

This correspondence presents an upper bound on the minimum distance of serially concatenated codes with interleaver where the inner code is a systematic recursive convolutional encoder and the outer code is any convolutional encoder. The resulting expression shows that the minimum distance of the concatenated code cannot grow more than O(K¹-1/d_f ^(O)), where K is the information word length, and d_f ^(O) is the free distance of the outer code. The obtained upper bound is shown to agree with and, in some cases, improve over previously known results