Bounds on the undetected error probabilities of linear codes for both error correction and detection

The (n, k , d 2 2t + 1) binary linear codes are studied, which are used for correcting error patterns of weight at most t and detecting other error patterns over a binary symmetric channel. In particular, for t = 1, it is shown that there exists one code whose probability of undetected errors is upper bounded by (n + 1]2"-k n ] l when used on a binary symmetric channel with transition probability less than 2 / n . I . INTRODUCTION In pure ARQ systems, linear codes are used solely for detecting errors. Suppose that we apply linear codes to a binary symmetric channel (BSC) with transition probability p. It 11, pp. 78-79] has been proved that for each p with 0 I p I 1 , there exists an ( n , k ) binary linear code whose probability of undetected errors (PUDE) is upper bounded by 2 ( n k ) . Hamming codes and double error correcting primitive BCH codes [2 ] , [3 ] have been proved to satisfy the inequality if the transition probability p is no greater than 1 / 2 . Pure ARQ systems have the problem of low throughput if the transition probability in the BSC is high. Therefore, in hybrid ARQ systems [ l ] especially in type-I hybrid ARQ systems, linear codes are used for correcting some low weight error patterns and detecting many other error patterns. Therefore, it is interesting to study the probability of undetected errors for linear codes that are used for both error correction and error detection over the BSC. In this correspondence, our study is divided into two parts. In the first part, we study the class of ( n , k , d 2 3) systematic linear codes that can be used for correcting every single error and detecting other error patterns. We show that there exists one code whose PUDE is upper bounded by ( n + 1) . [ 2 n k n]-l when the transition probability is less than 2 / n . In the second part, we study the ( n , k ) systematic linear codes that are used for correcting some low weight-error patterns and detecting other error patterns. Suppose that 1 R > H(2A). We show that there exists an ( n , Rn,d 2 2An + 1) linear code whose PUDE is closely upper bounded by 2 [ ' R H ' A ) ] n as n approaches infinity and the transition probability is less than A (if it is used to correct all the error patterns of weight at most An and to detect other error patterns). Manuscript received February 8, 1989; revised December 1, 1989. This work was presented at the IEEE 1990 International Symposium on Information Theory, San Diego, CA, January 14-19, 1990. This work was supported by the National Science Council of the Republic of China under grant NSC 78-0404-E002-05. M.-C. Lin is with the Department of Electrical Engineering, National Taiwan University, Taipei 10764, Taiwan, ROC. IEEE Log Number 9036388. 11. CODES FOR ERROR DETECTION A N D SINGLE-ERROR CORRECTION Consider the ensemble r of all systematic ( n , k , d 2 3) binary linear codes. The generator matrix of an ( n , k ) systematic linear code V is of the form G = [ I PI, where I is the k X k identity matrix and P is some k ( n k ) matrix. A necessary and sufficient condition for V to have minimum distance of at least 3 is that no two rows of P are identical and each row in P must have weight of at least 2. Therefore, the cardinality of r is iri = [ 2" k 1 ( k ) ] . [ 2" k 1 (. k ) 11 . . . [2"-k 1 ( n k) ( k l)] [ 2" 1 ( n k )] ! [ 2 n 4 1 n ] ! (1) . We denote the codes in r by VI, V , ; . .,Tr,. Let A l . , be the number of weight-w codewords in v, where I = 1,2; . ., Irl, and w = 0,3,4,. . . , n. Suppose v is used to correct every single error and detect other error patterns over a BSC with transition probability p , its PUDE is n JTEIv) = c [(w + l ) . A , , , + I + A I , , + ( n w + l ) . ~ , , w l ] .PW(1 p ) " W . (2) w = 2 If the probability of choosing each code in r is equally likely, the average PUDE over all the codes in r is Note that each nonzero n-tuple appears in at most Ir'l codes in r, where ir'i I [2n-k I ( ~ k ) ] I ( ~ k ) i ] . . . [2n-k 1 ( n k ) ( k -2)] 12n-k 1 ( n k ) l ! (4) 0018-9448/90/0900-1139$01.00 01990 IEEE 1140 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5 , SEPTEMBER 1990 with codes in rD, then f m = 2 c ,e ( ; ) ( : ) ( ; : : ) Combining (4)-(7) we have w = D m = O i = O n + l ( 1 p ) n w m + 2 i . P W + m 2r (8) ir'i iri P ( E ) I ( n + l ) I 2"-k. It follows from (8) that, for each p , there exists a code in r whose PUDE is at most n + 1 / 2 " k n. Note that the term ~"'(1P I " " is an increasing function of p if p I w / n . Hence, for each code y in I', P ( E I y ) is an increasing function of p if p I 2 / n . Therefore, we see that there exists at least one code in r such that its PUDE is upper bounded by 111. CODES FOR ERROR DETECTION A N D MULTIPLE-ERROR CORREC~ION The ensemble of all the systematic ( n , k ) linear codes contains 2 k ( n k ) distinct codes while at most of them contain nonzero codewords of weight less than d . Thus, the ensemble of all the systematic ( n , k , d 2 D ) linear codes rD contains distinct codes. Let V, be a code in r,, and let A/ , , , be the number of codewords of weight w in 5, where 1 = 1,2; . ., lrDl. Assume D = 2t + 1 . If V, is used for correcting all the error patterns of weight no more than t and detecting other error patterns, then its PUDE [4] is n I m i n ( t i , n w ) P(EIV,)= c 4 . w c c w = D i = O j = O If we define as zero for i > n or i < 0, then we can replace the index term of min(t i , n w) in (10) by t i . If each code in rD is selected equally likely, by taking the average of (10) over (3 = w = D m = O 5 i(;)E: i = O