IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-33, NO. 1, JANUARY, 1987 91 Encoding and Decoding for the Minimization of Message Symbol Error Rates in Linear Block Codes LARRY A. DUNNING, MEMBER, IEEE Abstract-Given any fixed linear block code, the error rates for the message symbols depend both on the encoding function and on the decoding map. This research shows how to optimize the choice of a generator matrix and decoding map simultaneously to minimize the error rates for all message symbols. The model used assumes that the distibu- tion of messages is flat and that the distribution of error vectors defining the channel is independent of the message transmitted. In addition, it is shown that, with proper choice of coset leaders, standard array decoding is optimal in this circumstance. The results generalize previously known results on unequal error protection and are sufficiently general to apply when a code is used for error detection only. I. INTRODUCTION T HE EFFECTIVENESS of error-correcting codes is frequently gauged by the probability of word error. In some applications, however, the bit error rates of the individual message positions are important. For example, several different data fields may be packaged into the same codeword. In such cases,the probability of bit or symbol error is often of importance [3], [21], [27]. The decoding method can influence this rate [l], [7], [12], [15], [16], [26] as can the encoding method when the symbol error rate for the message symbols is considered[27], [28]. In many applications the error rates of the individual messagesymbols are important and not just the average message symbol error rate. This occurs, for example, when numeric data are being transmitted and one wishes to minimize the mean-square error rate [4]-[6], [23]-[25], [33]. In this case lower error rates for message symbols corre- sponding to the most significant digits would be desirable. Other applications may package several data fields in a single codeword, and these fields may require differing degrees of protection. Such situations occur regularly when dealing with telemetry data and with the command and control of satellites. An analog of minimum distance un- equal error protection [2], [8]-[lo], [20], [29]-[31] has previ- ously been studied for use when such fields require differ- ent degrees of protection. This research focuses on the case where the individual error rates for the message symbols are used as a measureof the protection provided for such Manuscript received September 24, 1985; revised February 24, 1986. This paper was presented in part at the IEEE International Symposium on Information Theory, Brighton, England, June 23-28, 1985. The author is with the Department of Computer Science, Bowling Green State University, Bowling Green, OH 43403-0214. IEEE Log Number 8610110. fields. The results in [24] are also of interest in this connection. The development which follows studies the problem of minimizing the susceptibility of the message symbols to error in fixed linear block codes. Thus it is the encoding and decoding maps that are to be varied. The methods developed will give an encoding and decodingpair offering the greatest protection to the first messagesymbol and decreasing levels of protection to subsequent symbols in the message. The user may then rearrangethe order of his message symbols in order of importance to take advantage of this scheme. The results will be sufficiently general to allow the choice of any of several different measuresfor this susceptibility. The probability of symbol error when a code is used for error detection only and unequal error protection are included among these measures. The mini- mization of the average message symbol error rate with a fixed choice of coset leaders, as developedin [28], is also a special caseof theseresults. The focus, however, will be on developing an encoding generator matrix and a decoding map which, when used as a pair with complete decoding, will simultaneously minimize the error rates for all the message symbols. These mappings will also, of course, minimize the average message symbol error rate. In the next section, a procedure for finding encodings which are optimal with respect to a very general weight function will be given. Subsequently,Section III will show how to apply the procedurewhen a systematic encoding is required. Section IV will develop a special weight function for use in jointly choosing a generator matrix and coset leaders to minimize symbol error rates. Theorem 5 of Section IV summarizes this and constitutes the main result. The results of Section II, or Section III for systematic encoding, may then be used to find an optimal generator matrix. Section IV will then show how to compute the corresponding coset leaders.Section V applies theseresults to a sample code. Investigation of the sample code shows, for example, that minimizing the messagesymbol error rates may result in a higher word error rate. II. OPTIMALGENERATORMATRICES Consider the situation where an (n, k) linear block code C over GF(q) is used for error detection. We wish to .._ determine the probability that an undetected error will 1.00 01987 IEEE 0018-9448/87/0100-0091$0