Abstract Algebra Theory and Applications - Computer Science ...

132 CHAPTER 7 ALGEBRAIC CODING THEORYExercises1. Why is the following encoding scheme not acceptable?Information: 0 1 2 3 4 5 6 7 8Codeword: 000 001 010 011 101 110 111 000 0012. Without doing any addition, explain why the following set of 4-tuples in Z 4 2cannot be a group code.(0110) (1001) (1010) (1100)3. Compute the Hamming distances between the following pairs of n-tuples.(a) (011010), (011100)(c) (00110), (01111)4. Compute the weights of the following n-tuples.(a) (011010)(c) (01111)(b) (11110101), (01010100)(d) (1001), (0111)(b) (11110101)(d) (1011)5. Suppose that a linear code C has a minimum weight of 7. What are theerror-detection and error-correction capabilities of C?6. In each of the following codes, what is the minimum distance for the code?What is the best situation we might hope for in connection with error detectionand error correction?(a) (011010) (011100) (110111) (110000)(b) (011100) (011011) (111011) (100011)(000000) (010101) (110100) (110011)(c) (000000) (011100) (110101) (110001)(d) (0110110) (0111100) (1110000) (1111111)(1001001) (1000011) (0001111) (0000000)7. Compute the null space of each of the following matrices. What type of (n, k)-block codes are the null spaces? Can you find a matrix (not necessarily astandard generator matrix) that generates each code? Are your generatormatrices unique?(a)⎛⎝0 1 0 0 01 0 1 0 11 0 0 1 0⎞⎠(b)⎛⎜⎝1 0 1 0 0 01 1 0 1 0 00 1 0 0 1 01 1 0 0 0 1⎞⎟⎠