Assignment on Cyclic Codes
EE512: Error Control Coding
Questions marked (Q) or (F) are questions from previous quizzes or final exams,
respectively.
1. What is the ideal describing the cyclic code
{
0000
,
0101
,
1010
,
1111
}
?
2. Describe the smallest cyclic code containing the vector 0011010.
3. Show that in an (
n, k
) cyclic code any
k
consecutive bits can be taken to be the message bits.
4. Consider the
n
= 7 binary cyclic code generated by
g
(
x
) = 1 +
x
+
x
3
.
(a) Find all codewords of the code.
(b) The allzero codeword
c
(
x
) = 0 is obtained uniquely by multiplying
g
(
x
) by
m
(
x
) = 0 in
GF(2)[
x
]. Find all
f
(
x
)
∈
GF(2)[
x
]
/
(
x
7
+ 1) such that
f
(
x
)
g
(
x
) = 0 in GF(2)[
x
]
/
(
x
7
+ 1).
5. A binary cyclic code of length 15 has generator polynomial
g
(
x
) = (
x
4
+
x
+1)(
x
4
+
x
3
+
x
2
+
x
+1).
Give a generator matrix and paritycheck matrix for the code. Find the generator matrix for the
dual of the code.
6. Find the dimension and generator polynomial for every binary cyclic code of length 15, 17, 21, 31,
51, 73, 85.
7. Let
C
be the
n
= 3 cyclic code over GF(4)=
{
0
,
1
, α, α
2
}
(
α
3
= 1,
α
2
= 1 +
α
) generated by
g
(
x
) =
x
+
α
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 profnaganagi
 Coding theory, 1 g, 0 g, CYCLIC CODES, Cyclic code, binary cyclic code

Click to edit the document details