Coding Theory (CS 425)
Tutorial- I
January 29, 2025
1. Argue that the triplet ({0, 1}, +, ×) is a Field. Note that + is addition
(mod 2), same as binary ⊕ and × is multiplication (mod 2), same as binary
· (and operator).
2. Does ({0, 1, . . . , 6}, +6 , ×6 ) (denoted as Z6 ) constitute a field? What about
Z7 ?
3. With reference to the definitions of addition and multiplication in Qn. 1,
determine the product of the matrices:


0 1 0
1 0 0
0 1 1 0

×
1 1 1
1 0 1 1
0 1 1
4. Let

1
G = 0
0
0
1
0
0
0
1

0
0
1
be the generator matrix for a code. Find the codeword corresponding to
the message (1, 1, 1) and (1, 0, 0, ).
5. What is the generator matrix for the repetition code [3k, k, 3]2 ?
6. Consider [127, −, 3]2 Hamming code. Determine the dimensions of the
Parity Generator Matrix (P GM ) G and the Parity Check Matrix (P CM )
H. Further, argue that GH = 0
7. Argue that [2r − 1, r, 2r−1 ]2 Hadamard code is linear.
8. Identify the minimal extension (add as few codewords as possible) of the
code {100, 001, 101} so as to make it linear?
9. Construct a linear code (the set of codewords) over F3 of length (n) 5 and
dimension (k) 2 where the distance of the code is 2 (Note that F3 is a field
with addition and multiplication modulo 3).
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10. Consider a set F = {(0, 0), (0, 1), (1, 0), (1, 1)}. Define addition and multiplication on F as coordinatewise addition and multiplication modulo 2.
Is F a field under these operations?
11. Consider the set of all polynomials of degree at most 1 with coefficients
from F2 . Let addition and multiplication be modulo x2 + x + 1. Check
whether the given structure forms a field or not. List the addition and
multiplication tables if so.
12. Assume that there exist an (n, k, d)q code C over Fq :
(a) Then there exists (n + r, k, d)q code for every r ≥ 1.
(b) Then there exists (n − r, k, d − r)q code for every r = 1, 2, · · · d − 1
(c) Then there exists (n − r, k − r, d)q code for every r = 1, 2, · · · k − 1
13. Let C1 , C2 be linear codes such that for i ∈ {1, 2}, Ci is an [ni , ki , di ]q .
Define the direct sum
C = {c1 c2 |c1 ∈ C1 , c2 ∈ C2 }
is an (n1 +n2 , k1 +k2 , min(d1 , d2 ))q code. Write down the generator matrix
for the direct sum code.
14. Let C1 , C2 be linear codes such that for i ∈ {1, 2} [n.ki , di ]q . Define the direct sum C = {(c1 , c1 +c2 )|c1 ∈ C1 , c2 ∈ C2 } is a [2n, k1 +k2 , min(2d1 , d2 )]q
code.
15. Let x be bitwise complement of x and let C be an [n, k, d]2 code. Show
that
C ′ = {cc|c ∈ C} ∪ {cc|c ∈ C}
is a [2n, k + 1, min(2d, n)]2 code.
16. Consider a [15, 4, 6]3 code. What is the volume of (number of words inside)
a hamming ball of radius 3?
17. In a binary linear code, either all codewords begin with a 0 (or 1) or
exactly half of the codewords begin with a zero.
18. Let G be a generator matrix of an (n, k, d)2 binary linear code. Then show
that G has at least kd ones in it.
19. Find the quotient and remainder upon dividing f (x) = x4 − 3x3 + 2x2 +
4x − 1 by g(x) = x2 − 2x + 3 in Z5 [x].
20. Find all zeros/roots of f (x) = 3x2 + x + 4 over Z7 .
21. Find the [4, ]2 Cyclic code generated by the polynomial 1+x2 . Determine
the corresponding parity generator matrix and parity check matrix.
22. Argue that the polynomial g(x) of minimal degree in a cyclic code:
2
(a) Divides every other codeword polynomial
(b) Divides (xn − 1)
23. Check whether F2 [x]/ < x2 + x + 1 > is a field or not (polynomials, whose
coefficients are {0, 1} and addition and multiplication modulo x2 + x + 1).
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