University of California San Diego
ECE 259A:
Solutions to Problem Set #4
1
. A binary polynomial
g x
generates a linear cyclic code of length 15 if and only if it is a divisor
of
x
15
1
over
GF 2
. Using the factorization
x
15
1
x
1
x
2
x
1
x
4
x
1
x
4
x
3
1
x
4
x
3
x
2
x
1
we conclude that there are
2
5
32
binary linear cyclic codes of length 15, including the trivial
cyclic codes
0
and
IF
n
2
. Note that the factorization of
x
15
1
itself is not essential to answer
this question. All one needs to know is the total number of factors, which may be easily found
by writing out the cyclotomic cosets.
2
.
(a)
If
x
1
is a factor of
g x
then
g
1
0
, and thereforee for all
c x
, we have
c
1
c
0
c
1
c
n
1
f
1
g
1
0
.
Since
g x
, we have
g
0
g
1
g
r
g
1
0
which means that
1
α
0
is a root of
g x
. Therefore, the minimal polynomial of
α
0
div
ides
g x
. Yet the minimal polynomial of
α
0
is
x
1
over any field.
(b) The allone vector is described by the polynomial
1
x
x
n
1
x
n
2
x
1
. In
any field we have
x
n
1
x
1
1
x
. Since
n
and
q
are relatively prime, all the roots
of
x
n
1
are distinct and therefore
1
α
0
is not a root of
1
x
. Thus if
1
x
then
α
0
cannot be a root of
g x
. On the other hand, since all the nonzero powers of
α
are
roots of
1
x
, if
α
0
is not a root of
g x
then necessarily
g x
1
x
and hence
1
x
.
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 Fall '08
 Prof.AlexanderVardy
 Polynomials, Cyclic group, Polynomial code, Cyclic code

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