# hw-4-2009 - A Range Problems 1 • 4 Roth text problem 4.14...

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ECE 556: Coding Theory, 2009 HW 4 Issued: October 5th, 2009 Due: October 15th, 2009 October 5, 2009 A- range Problems 1) Show that over any ﬁnite ﬁeld of order q and characteristic p , one has ( x + y ) p = x p + y p . (1) 2) Roth text, problem 4.29. 3) Consider the code with H matrix given by: H = [ H 1 I 3 × 3 ] , (2) where H 1 is the parity check matrix of the [7 , 4 , 3] Hamming code. Give an interpretation of this code, i.e. give a shorthand description of its codewords, and show how this leads to the weight enumerator quickly. To calculate the exact probability of error for signaling, assuming that we have a ML decoder and that the transmission is over a BSC, we must know the weights of the coset leaders. Find these as well. Remark : Do not use a computer to solve this problem.

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Unformatted text preview: A Range Problems 1 • 4) Roth text, problem 4.14. You are allowed to search online for perfect ternary codes. • 5) SVHITDXEHOERCPRUOEERASTNPBMOHTFTLTFSNIETRLFMAEO The space symbols are not encrypted. A+ Range Problems • 6) The covering radius of a code is the largest weight of a coset leader of the code. What can you say about the covering radius of perfect codes? Can you ﬁnd an upper bound on the covering radius of an arbitrary linear block code? The covering radius of codes that meet the Singleton bound, Griesmer bound (i.e. can you tell if there is any connection at all between the covering radius and the bounds at hand)? 2...
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## This note was uploaded on 02/08/2012 for the course ECE 556 taught by Professor Milenkovic,o during the Fall '08 term at University of Illinois, Urbana Champaign.

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hw-4-2009 - A Range Problems 1 • 4 Roth text problem 4.14...

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