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Instructions. Name:
Exam Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. Put your name on each page of your paper. 1. [25 Points] (a) Fill in the missing parts to complete the statement of Burnside's Theorem: If G is a finite group that acts on a finite set X, then the number N of distinct orbits is: 1 N= .
gG
(b) A wheel is divided evenly into 4 different compartments. Each compartment can be painted red (R), white (W), blue (B) or green (G). The back of the wheel is black. We consider two color patterns the same provided one can be obtained from the other by means of a rotation of 0 , 90 , 180 , or 270 . i. The following picture describes one color pattern. Draw all of the color patterns that are equivalent to this one. That is, the ones that are obtained from this one by means of all of the rotations. W G R W
ii. How many different 4 compartment color wheels are there under the above assumptions of (1) 4 possible colors and (2) wheels are considered the same if one is obtained another by means of a rotation? 2. [18 Points] Suppose that C is a binary (23, 12, 7) code. Answer the following questions C. about (Recall that a linear code is (n, k, d) if n is the word length, k is the dimension, and d is the minimum distance.) (a) What is |C|? (That is, how many code words are there?) (b) What is the size of a generator matrix for C? (c) What is the size of a parity check matrix for C? (d) How many error's can C detect? (e) How many error's can C correct? (f) Let x = 10001000100010001000100. Explain why x could not possibly be a codeword. ^ ^
Math 4023 Section 1
April 24, 2009
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Name:
Exam 3
3. [32 Points] Let C be the binary code with 1 0 G = 0 1 0 0 (a) List all of the codewords of C. (b) Give a parity check matrix for C. (c) What is the minimum distance d of C? (d) Compute a syndrome table for C.
generator matrix 0 1 0 1 0 1 1 1 . 1 0 1 1
(e) ...