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Coding Theory – Winter 2022 361-2-6251 ECE, BGU

Coding Theory – Winter 2022 361-2-6251 ECE, BGU

 

Due date: 20/12/2021 at 23:59 You should answer all questions. You can use the claims of previous questions to prove later ones (even if you do not know the answer to the previous questions). All claims should be proved. All calculations should be explained. The homework is to be solved alone without consulting with other people. Other sources are allowed, but should be clearly cited.

 

Homework #2

 

1. 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

 

αri = 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 βq = β.

 

2. 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′ linear subcode of C

 

dim(C′)=`

 

∣∣supp(C′)∣∣ . (a) (10%) Prove d1(C) = d.

 

(b) (15%) Prove that if C′ is an [n, k′] subcode of C with k′ > 1, there exists a code C′′ which is an [n, k′ − 1] subcode of C′ such that |supp(C′′)| < |supp(C′)|.

 

(c) (15%) Prove for all 1 6 ` 6 k − 1 that d`(C) < d`+1(C).

 

3. 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.

 

4. (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

 

ᵀ 1 , . . . , h

 

ᵀ n. Let r(C) be defined as in Question 3 of Homework 1. Assume ` is the smallest

 

integer such that for any column vector vᵀ of length n − k there exist ` columns of H, hᵀi1 , . . . , h ᵀ i` , and ` scalars

 

α1 , . . . , α` ∈ Fq such that vᵀ = ∑`j=1 αj h ᵀ i j . Prove r(C) = `.

 

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