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Problem Set 2 for ENEE626 (2007) - Galois Theory and Finite Fields - Prof. Alexander Barg, Assignments of Electrical and Electronics Engineering

University of MarylandElectrical and Electronics Engineering
Prof. Alexander Barg

Problem set 2 for the enee626 (2007) course on galois theory and finite fields. The problems involve working with finite fields, irreducible polynomials, and extensions of fields. Students are asked to show that certain polynomials are not primitive, construct finite fields with given polynomials, find the order and basis of roots, and determine the number of primitive elements in a finite field. Other problems involve working with reed-solomon codes.

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Pre 2010

Uploaded on 02/13/2009

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ENEE626 (2007). Problem set 2. Due in class on 10/18/07.
1. Take as a given that the polynomial f(x) = x4+x3+x2+x+ 1 is irreducible over F2.
(a) Show that fis not primitive.
(b) Construct F16 by adding a root βof fto F2. More concretely, in every row of the table of F16 write
the coefficients of the expansion of the corresponding element into the basis 1, β, β2, β 3.
2. Construct F16 as an extension of F4.Namely, do the following:
Let αbe the primitive element of F16 that satisfies α4=α+ 1 (refer to the table of F16 from the class
notes).
(a) Let F4={0,1, ω, ¯ω}.Using this notation, write out the multiplication and addition tables in F4.Find
isuch that ω=αi,find jsuch that ¯ω=αj.
(b) Prove that f(x) = x2+ωx + 1 is irreducible over F4.
(c) Let βbe a root of f(x). What is the order of β? Is βprimitive?
(d) Let β=αi.What is i?
(e) Prove that (β, 1) form a basis of F16 over F4.Write out coefficients of the expansion of every element
in F16 in this basis (in other words, write a representation of every element of F16 as a polynomial over F4).
(f) Find all ireducible polynomials of degree ≤2over F4of the form x2+ax +b.
3. (a) Determine the number of primitive elements of F32.
(b) Show that the polynomial f(x) = x5+x2+ 1 is irreducible over F2.
(c) Are there elements γ∈F32 of order 15?
Let αbe a zero of f(x).
(d) Compute Q4
i=0(x−αi).
(e) Compute the logarithm of α4+α3+α.
(f) Let γ∈F32\F2.Show that γis not a root of a polynomial of degree less than 5.
(g) Show that 1, γ, γ2, γ 3, γ4is a basis for F32 as a linear space over F2.
(h) What are the coordinates of α8with respect to the basis 1, α, α2, α3, α4?
4. (a) Prove that α= 2 is a primitive element of F=F11.
(b) Let Cbe a [n= 10, k = 6] RS code over F. Write out a parity-check matrix Hof F.
(c) Reduce Hto a systematic form H0= [I4|A].
(d) Write out the generator matrix of Cin a systematic form G= [I5|B].
(e) Using H0,find the codeword c0that corresponds to the message symbols (1,1,1,1,1,1).
(f) What is the polynomial fsuch that eval(f) = c0?
(g) Let r= (3,0,0,10,5,4,0,6,10,0) be a received vector. Decode it using the Berlekamp-Welch
algorithm, show the steps.

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ENEE626 (2007). Problem set 2. Due in class on 10/18/.

1. Take as a given that the polynomial f (x) = x^4 + x^3 + x^2 + x + 1 is irreducible over F 2. (a) Show that f is not primitive. (b) Construct F 16 by adding a root β of f to F 2. More concretely, in every row of the table of F 16 write the coefficients of the expansion of the corresponding element into the basis 1 , β, β^2 , β^3. 2. Construct F 16 as an extension of F 4. Namely, do the following: Let α be the primitive element of F 16 that satisfies α^4 = α + 1 (refer to the table of F 16 from the class notes). (a) Let F 4 = { 0 , 1 , ω, ω¯}. Using this notation, write out the multiplication and addition tables in F 4. Find i such that ω = αi, find j such that ¯ω = αj^. (b) Prove that f (x) = x^2 + ωx + 1 is irreducible over F 4. (c) Let β be a root of f (x). What is the order of β? Is β primitive? (d) Let β = αi. What is i? (e) Prove that (β, 1) form a basis of F 16 over F 4. Write out coefficients of the expansion of every element in F 16 in this basis (in other words, write a representation of every element of F 16 as a polynomial over F 4 ). (f) Find all ireducible polynomials of degree ≤ 2 over F 4 of the form x^2 + ax + b. 3. (a) Determine the number of primitive elements of F 32. (b) Show that the polynomial f (x) = x^5 + x^2 + 1 is irreducible over F 2. (c) Are there elements γ ∈ F 32 of order 15? Let α be a zero of f (x). (d) Compute ∏^4 i=0(x − αi). (e) Compute the logarithm of α^4 + α^3 + α. (f) Let γ ∈ F 32 \F 2. Show that γ is not a root of a polynomial of degree less than 5. (g) Show that 1 , γ, γ^2 , γ^3 , γ^4 is a basis for F 32 as a linear space over F 2. (h) What are the coordinates of α^8 with respect to the basis 1 , α, α^2 , α^3 , α^4? 4. (a) Prove that α = 2 is a primitive element of F = F 11. (b) Let C be a [n = 10, k = 6] RS code over F. Write out a parity-check matrix H of F. (c) Reduce H to a systematic form H′^ = [I 4 |A]. (d) Write out the generator matrix of C in a systematic form G = [I 5 |B]. (e) Using H′, find the codeword c 0 that corresponds to the message symbols (1, 1 , 1 , 1 , 1 , 1). (f) What is the polynomial f such that eval (f ) = c 0? (g) Let r = (3, 0 , 0 , 10 , 5 , 4 , 0 , 6 , 10 , 0) be a received vector. Decode it using the Berlekamp-Welch algorithm, show the steps.