 En < 1 (or K - 1 > 0), substituting the inequality in Eq. (I11) into that in Eq. (H5) to obtain the final result: (I13) 2 T.M. Cover and J.A. Thomas, Elements of Information Theory (New York: John Wiley & Sons, 1991), p. 320-1. Lempel-Ziv distinct parsing Proof of Eq. (I3) Setting q = 2, we write the sum in Eq. (I2) in the form Lk=~mq" = =92mqm Since the sum involved in the last right-hand side is equal to (1 - q k+1)/(1-q) we -(K +1)q(1-q)-(1 -q+1)(-1) (1-q)2 = q[1-q*+++(k +1)q^ = Appendix J (Chapter 11) Error-correction capability of linear block codes In this appendix, I show that linear block codes have the capability of correcting any error patterns of Hamming weight w(E) = e, or any number of e errors in the received block code, provided that where dmin is the minimum Hamming distance, and with the brackets corresponding to the integer-floor definition, i.e., {(dmin - 1)/2} = L(dmin - 1)/2] corresponding to the highest integer contained in the argument. To show this, assume a block code C, with a minimum Hamming distance dmin. To recall, dmin represents the minimum number of bit positions for which two codewords differ in C. Let X be any codeword belonging to C, and Y any received block code, which may contain errors. Define the Hamming distance between vectors X and Y as d(X,Y). It is clear that if d(X, Y) < dmin, or equivalently, H(X,Y) < d min - 1, the block code Y does not belong to C. In this case, any possible error pattern is detected by the code. Assume next that X and Y are the transmitted and received vectors, respectively. Since the minimum Hamming distance dmin is either an odd or an even integer, we can write 2t+1<dmin<2+2, where t > 0 is an integer. I am now going to show that the code is capable of correcting all error patterns having t or fewer errors. Let Z be another vector in C. Using the distance property referred to as triangle inequality (see Chapter 5), we have d(X,Z)<d(X,Y)+d(Y,Z). Suppose now that the number of errors in the received vector Y is w(E) = e. This means d(X,Y)=e. The f