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# hw-5-2009 - in the ﬁeld Find a primitive element of the...

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ECE 556: Coding Theory, 2009 HW 5 Issued: October 21st, 2009 Due: November 3rd, 2009 October 21, 2009 A- range Problems 1) A Hadamard matrix H n of order n is a an n × n amtrix of +1’s and - 1’s such that H n H T n = nI , where I denotes the identity matrix. If H n is a Hadamard matrix, verify that H 2 n = ± H n H n H n - H n ² (1) is also a Hadamard matrix of order 2 n . Repalce each +1 with 0, and - 1 with 1. If the code consists of the rows of H n and their complements, what kind of code is it? What are its properties, parameters? 2) For the ﬁeld F 32 , describe all elements of the ﬁeld in polynomial form, using an irreducible polynomial of degree ﬁve to generate the ﬁeld. Give an example of how to ﬁnd the inverse element of an arbitrary element

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Unformatted text preview: in the ﬁeld. Find a primitive element of the ﬁeld, and write the representation of each ﬁeld element using the primitive element. Also, ﬁnd all the cyclotomic cosets of the ﬁeld, as well as all minimal polynomials of the ﬁeld element. • 3) Roth text, problem 3.21. 1 • 4) Roth text, problem 3.22. • 5) Find the number of information bits in a binary, narrow-sense BCH code of length 63 for each odd t satisfying 1 ≤ t ≤ 31. Find the a formula for the exact number of information symbols in a three error-correctig, primitive, narrow-sense BCH code for all m ≥ 4, where n = 2 m-1. 2...
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hw-5-2009 - in the ﬁeld Find a primitive element of the...

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