Linear Unequal Error Protection Research Articles

Linear unequal error protection codes based on terminated convolutional codes

Convolutional codes which are terminated by direct truncation (DT) and zero tail termination provide unequal error protection. When DT terminated convolutional codes are used to encode short messages, they have interesting error protection properties. Such codes match the significance of the output bits of common quantizers and therefore lead to a low mean square error (MSE) when they are used to encode quantizer outputs which are transmitted via a noisy digital communication system. A code construction method that allows adapting the code to the channel is introduced, which is based on time-varying convolutional codes. We can show by simulations that DT terminated convolutional codes lead to a lower MSE than standard block codes for all channel conditions. Furthermore, we develop an MSE approximation which is based on an upper bound on the error probability per information bit. By means of this MSE approximation, we compare the convolutional codes to linear unequal error protection code construction methods from the literature for code dimensions which are relevant in analog to digital conversion systems. In numerous situations, the DT terminated convolutional codes have the lowest MSE among all codes.

Constructing linear unequal error protection codes from algebraic curves

We show that the concept of generalized algebraic geometry codes which was introduced by Xing, Niederreiter, and Lam (see ibid., vol.45, p.2498-2501, Nov. 1999) gives a natural framework for constructing linear unequal error protection codes.

Soft-combining technique for LUEP codes

Decoding of linear unequal error protection (LUEP) codes using soft-combining is presented. Codewords are repeated and combined until the desired reliability is obtained. The decoding is performed in the spectral domain, offering reduced implementation complexity. Simulations show a significant performance improvement with soft-combining compared to soft decision decoding.

On block-coded modulation using unequal error protection codes over Rayleigh-fading channels

This paper considers block-coded 8-phase-shift-keying (PSK) modulations for the unequal error protection (UEP) of information transmitted over Rayleigh-fading channels. Both conventional linear block codes and linear UEP (LUEP) codes are combined with a naturally labeled 8-PSK signal set, using the multilevel construction of Imai and Hirakawa (1977). Computer simulation results are presented showing that, over Rayleigh-fading channels, it is possible to improve the coding gain for the most significant bits with the use of binary LUEP codes as constituent codes, in comparison with using conventional binary linear codes alone.

QPSK block-modulation codes for unequal error protection

Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u~|u~+v~|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Modified generalized concatenated codes and their application to the construction and decoding of LUEP codes

Proposes a modification of generalized concatenated codes, which allows to construct some of the best known binary codes in a simple way. Furthermore, a large class of optimal linear unequal error protection codes (LUEP codes) can easily be generated. All constructed codes can be efficiently decoded by the Blokh-Zyablov-Zinov'ev algorithm if an appropriate metric is used. >

On a class of optimal nonbinary linear unequal-error-protection codes for two sets of messages

Several authors have addressed the problem of designing good linear unequal error protection (LUEP) codes. However, very little is known about good nonbinary LUEP codes. The authors present a class of optimal nonbinary LUEP codes for two different sets of messages. By combining t-error-correcting Reed-Solomon (RS) codes and shortened nonbinary Hamming codes, they obtain nonbinary LUEP codes that protect one set of messages against any t or fewer symbol errors and the remaining set of messages against any single symbol error. For t/spl ges/2, they show that these codes are optimal in the sense of achieving the Hamming lower bound on the number of redundant symbols of a nonbinary LUEP code with the same parameters.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Binary coding by integration of polynomials

Based on the idea of integration and differentiation of polynomials, a large class of linear unequal error-protection (LUEP) codes is constructed. Many of these codes are optimal. A codeword is generated by binary discrete integration of an all-zero vector, using the information bits as integration constants. Decoding is performed by discrete differentiation of the received word. For special designs, all information bits are equally protected and in this case all classes of Reed-Muller codes are obtained. Thus, a new and very comprehensive description of these codes is given. The described codes are decoded by majority decisions over corresponding derivatives, based on the structure of Pascal's triangle reduced modulo 2. >

Nonlinear unequal error-protection codes are sometimes better than linear ones

The parameters of some binary nonlinear two-level unequal error-protection (UEP) codes are evaluated and compared with nonexistence bounds for linear UEP codes. The major points is that for UEP codes linearity is an essential restriction. The results demonstrate that at least in certain cases nonlinear codes perform better than linear ones. This is in contrast to the situation for ordinary codes, when in most cases the best possible result can be obtained even with the restriction of linearity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Codes with multi-level error-correcting capabilities

In conventional channel coding, all the information symbols of a message are regarded equally significant, and hence codes are devised to provide equal protection for each information symbol against channel errors. However, in some circumstances, some information symbols in a message are more significant than the other symbols. As a result, it is desirable to devise codes with multi-level error-correcting capabilities. In this paper, we investigate block codes with multi-level error-correcting capabilities, which are also known as unequal error protection (UEP) codes. Several classes of UEP codes are constructed. One class of codes satisfies the Hamming bound on the number of parity-check symbols for systematic linear UEP codes and hence is optimal.

On decoding unequal error protection product codes

The decoding of unequal error protection product codes, which are a combination of linear unequal error protection (UEP) codes and product codes, is addressed. A nonconstructive proof of the existence of a good error-erasure-decoding algorithm is presented; however, obtaining the decoding procedure is still an open research problem. A particular subclass of UEP product codes is considered, including a decoding algorithm that is an extension of the Blokh-Zyablov decoding algorithm for product codes. For this particular subclass the decoding problem is solved.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Extension of the BCH decoding algorithm to decode binary cyclic codes up to their maximum error correction capacities

The BCH algorithm is extended to correct more errors than indicated by the BCH bound. In the first step of the decoding procedure, a number of errors corresponding to a particular case of the Hartmann-Tzeng bound are corrected. In the second step full error correction is the goal. A measure for the worst-case number of field elements of an extension field GF(2/sup m/) that must be tested for this purpose is given for binary cyclic linear unequal error protection codes as well as for conventional binary cyclic codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Linear unequal error protection codes from shorter codes (Corresp.)

The methods for combining codes, such as the direct sum, direct product, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|u|u + v|</tex> constructions, concatenation, etc., are extended to linear unequal error protection codes.

Two topics on linear unequal error protection codes: Bounds on their length and cyclic code classes

It is possible for a linear block code to provide more protection for selected positions in the input message words than is guaranteed by the minimum distance of the code. Linear codes having this property are called linear unequal error protection (LUEP) codes. Bounds on the length of a LUEP code that ensures a given unequal error protection are derived. A majority decoding method for certain classes of cyclic binary UEP codes is treated. A list of short (i.e., of length less than 16) binary LUEP codes of optimal (i.e., minimal) length and a list of all cyclic binary UEP codes of length less than 40 are included.

Linear unequal error protection codes

The properties of linear codes over GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> that provide unequal error protection (UEP) of information digits are discussed. A design is proposed for optimal binary systematic linear UEP codes. Broad classes of iterative and concatenated UEP codes are constructed. Majority decoding algorithms for linear iterative UEP codes are described.