Abstract Algebra Theory and Applications - Computer Science ...

136 CHAPTER 7 ALGEBRAIC CODING THEORYgenerates a Hamming code. What are the error-correcting propertiesof a Hamming code?(b) The column corresponding to the syndrome also marks the bit thatwas in error; that is, the ith column of the matrix is i written as abinary number, and the syndrome immediately tells us which bit is inerror. If the received word is (101011), compute the syndrome. In whichbit did the error occur in this case, and what codeword was originallytransmitted?(c) Give a binary matrix H for the Hamming code with six informationpositions and four check positions. What are the check positions andwhat are the information positions? Encode the messages (101101) and(001001). Decode the received words (0010000101) and (0000101100).What are the possible syndromes for this code?(d) What is the number of check bits and the number of information bitsin an (m, n)-block Hamming code? Give both an upper and a lowerbound on the number of information bits in terms of the number ofcheck bits. Hamming codes having the maximum possible number ofinformation bits with k check bits are called perfect. Every possiblesyndrome except 0 occurs as a column. If the number of informationbits is less than the maximum, then the code is called shortened. Inthis case, give an example showing that some syndromes can representmultiple errors.Programming ExercisesWrite a program to implement a (16, 12)-linear code. Your program should beable to encode and decode messages using coset decoding. Once your program iswritten, write a program to simulate a binary symmetric channel with transmissionnoise. Compare the results of your simulation with the theoretically predicted errorprobability.References and Suggested Readings[1] Blake, I. F. “Codes and Designs,” Mathematics Magazine 52 (1979), 81–95.[2] Hill, R. A First Course in Coding Theory. Oxford University Press, Oxford,1986.[3] Levinson, N. “Coding Theory: A Counterexample to G. H. Hardy’s Conceptionof Applied Mathematics,” American Mathematical Monthly 77 (1970),249–58.[4] Lidl, R. and Pilz, G. Applied Abstract Algebra. Springer-Verlag, New York,1984.