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Elements as powers of a primitive element.

Let F = Fq be a finite field.
(a) (10%) Let α1,α2, . . . ,αq be the elements of F = Fq, assuming now that q > 3, and let r be an integer,
1 6 r < q − 1. Show that
q
∑
i=1
α
r
i = 0.
Hint: Represent the elements as powers of a primitive element.
(b) (10%) Let Φ = Fqm be an extension of F = Fq. Let β ∈ Φ be any element. Prove that β ∈ F if and only if
β
q = β.
Let C be an [n, k, d] code over Fq. Denote by supp(C) the set of coordinates of C that are not always 0, namely,
supp(C) = {1 6 i 6 n : there exists c = (c1, . . . , cn) ∈ C such that ci 6= 0} .
We also define for all 1 6 6 k the following quantity: d
(C) = min
C
0
linear subcode of C
dim(C
0
)= supp(C 0 ) . (a) (10%) Prove d1(C) = d. (b) (15%) Prove that if C 0 is an [n, k 0 ] subcode of C with k 0 > 1, there exists a code C 00 which is an [n, k 0 − 1] subcode of C 0 such that |supp(C 00)| < |supp(C 0 )|. (c) (15%) Prove for all 1 6 6 k − 1 that d (C) < d+1
(C).
Let C be an [n, k, d] MDS code over Fq.
(a) (10%) Let G be any generating matrix for C. Prove that any subset of k columns of G forms a k × k invertible
matrix.
(b) (10%) Prove C
⊥ is also MDS. Hint: Use question 3a.
(20%) Let C be an [n, k] code over Fq, and let H be an (n − k) × n parity-check matrix for the code, whose
columns are denoted h
|
1
, . . . , h
|
n
. Let r(C) be defined as in Question 3 of Homework 1. Assume is the smallest integer such that for any column vector v | of length n − k there exist columns of H, h
|
i1
, . . . , h
|
i , and scalars
α1, . . . ,α∈ Fq such that v | = ∑
j=1αjh
|
i
j
. Prove r(C) = `.

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