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Unformatted text preview: EE 231E Spring 08 Midterm Channel Coding Tuesday, May 6, 2008 Instructor: Rick Wesel 82+10 pts, 110 minutes SOLUTIONS: 1. (10 pts) Proof of Singleton. State and prove the Singleton Bound: Solution: The Singleton bound states that d min n k + 1. The proof is as follows: For an alphabet of size q , there are q n possible sensewords. Consider a subword using k 1 of the n symbol positions. There are q k 1 possible subword values, but q k possible codewords. Hence at least two codewords share the same subword value. These two codewords differ in at most n ( k 1) symbol positions, proving the bound. 1 2. (10 pts) Galois Field 2 3 For GF (2 3 ), a primitive polynomial is x 3 + x + 1. Generate representations for the powers of as polynomials in that are degreetwo or less. Then, give the degree twoorless polynomial representation of 3 + 5 . j Polynomial in of degree less than 3 2 2 3 + 1 4 2 + 5 2 + + 1 6 2 + 1 7 1 3 + 5 = + 1 + 2 + + 1 (1) = 2 (2) 3. (10 pts) Factors and Blocklengths. (a) Which polynomials of the form x n 1 are factors (over GF (2)) of x 63 1? Solution: The values of n that are factors of 63 are the only ones that work since this implies the existence of an order n element. x 1 (3) x 3 1 (4) x 7 1 (5) x 9 1 (6) x 21 1 (7) x 63 1 (8) (b) What are the possible blocklengths of cyclic codes with symbols from GF (64)? Solution: For G ( x ) to generate a cyclic code of blocklength n it must be a factor of x n 1. This is only possible when x n 1 has all of its roots in the extension field. Hence the possible blocklengths are all factors of 63: { 1 , 3 , 7 , 9 , 21 , 63 } 2 4. (10 pts) BCH Code Design. The following table lists the conjugates and their corresponding minimal polynomials for GF(64). Table 1: Cyclotomic cosets (conjugates) and their minimal polynomials for GF(64). Conjugacy Class Minimal Polynomial 1 x + 1 , 2 , 4 , 8 , 16 , 32 x 6 + x + 1 3 , 6 , 12 , 24 , 33 , 48 x 6 + x 4 + x 2 + x + 1 5 , 10 , 17 , 20 , 34 , 40 x 6 + x 5 + x 2 + x + 1 7 , 14 , 28 , 35 , 49 , 56 x 6 + x 3 + 1 9 , 18 , 36 x 3 + x 2 + 1 11 , 22 , 25 , 37 , 44 , 50 x 6 + x 5 + x 3 + x 2 + 1 13 , 19 , 26 , 38 , 41 , 52 x 6 + x 4 + x 3 + x + 1 15 , 30 , 39 , 51 , 57 , 60 x 6 + x 5 + x 4 + x 2 + 1 21 , 42 x 2 + x + 1 23 , 29 , 43 , 46 , 53 , 58 x 6 + x 5 + x 4 + x + 1 27 , 45 , 54 x 3 + x + 1 31 , 47 , 55 , 59 , 61 , 62 x 6 + x 5 + 1 Design a binary BCH code with length n = 63 for design distance = 8. You may express the generator polynomial as a product of minimal polynomials. To get full credit you should find the highest rate code possible.credit you should find the highest rate code possible....
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This note was uploaded on 03/29/2011 for the course EECS 100 taught by Professor Dun during the Spring '11 term at CSU Channel Islands.
 Spring '11
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