North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH – 212 (B) (Computational Algebra) - university paper

many errors will e detect?
14
7) Consider (3 , 8) encoding function e : B3 ? B8 described by e(000) =
00000000, e(001) = 10111000, e(010) = 00101101, e(100) =
10100100, e(011) = 10010101, e(101) = 10001001, e(110) =
00011100, e(111) = 00110001. How many errors will e detect?
8) Consider (3 , 8) encoding function e : B3 ? B8 described by e(000) =
00000000, e(001) = 10111000, e(010) = 00101101, e(100) =
10100100, e(011) = 10010101, e(101) = 10001001, e(110) =
00011100, e(111) = 00110001. Is e a group code? Why?
9) Consider (2 , 6) encoding function e : B2 ? B6 described by e(00) =
000000, e(01) = 011110, e(10) = 101010, e(11) = 111000. obtain the
minimum distance of e. Is e a group code? Why?
10) Consider (2 , 6) encoding function e : B2 ? B6 described by e(00) =
000000, e(01) = 011110, e(10) = 101010, e(11) = 111000. How many
errors will e detect?
11) Let e be (3 , 5) encoding function described by e(000) = 00000, e(001) =
11110, e(010) = 01101, e(100) = 01010, e(011) = 10011, e(101) =
10100, e(110) = 00111, e(111) = 11001. Show that e is a group code.
12) Let e be (3 , 5) encoding function described by e(000) = 00000, e(001) =
11110, e(010) = 01101, e(100) = 01010, e(011) = 10011, e(101) =
10100, e(110) = 00111, e(111) = 11001. How many errors will e
detect?
13) Let H =
? ? ? ? ? ?
?
?
? ? ? ? ? ?
?
?
0 0 1
0 one 0
1 0 0
0 one 1
1 one 0
be a parity check matrix. Determine the group
code eH : B2 ? B5.
15
14) Let H =
? ? ? ? ? ? ? ?
?
?
? ? ? ? ? ? ? ?
?
?
0 0 1
0 one 0
1 0 0
0 one 1
1 0 1
1 one 0
be a parity check matrix. Determine the group code
eH : B3 ? B6.
15) Let H =
? ? ? ? ? ? ? ?
?
?
? ? ? ? ? ? ? ?
?
?
0 0 1
0 one 0
1 0 0
1 one 1
0 one 1
1 0 0
be a parity check matrix. Determine the group code
eH : B3 ? B6.
16) Consider (3 , 8) encoding function e : B3 ? B8 described by e(000) =
00000000, e(001) = 10111000, e(010) = 00101101, e(100) =
10100100, e(011) = 10010101, e(101) = 10001001, e(110) =
00011100, e(111) = 00110001. Let d be an (8 , 3) maximum
likelihood decoding function associate with e. How many errors can
(e,d) detect?
17) Consider (3 , 5) encoding function e : B3 ? B5 described by by e(000) =
00000, e(001) = 11110, e(010) = 01101, e(100) = 01010, e(011) =
10011, e(101) = 10100, e(110) = 00111, e(111) = 11001. Let d be an
(5 , 3) maximum likelihood decoding function associate with e. How
many errors can (e,d) detect?
18) Let e be the (3 , 8) encoding function with minimum distance 4. Let d
be the associated maximum likelihood decoding function. Determine
the number of errors that (e,d) can accurate.
16
19) obtain i)
? ? ? ?
?
?
? ? ? ?
?
?
1 one 0 1
0 one 1 0
0 one 1 0
1 0 0 1
?
? ? ? ?
?
?
? ? ? ?
?
?
1 one 0 1
0 one 1 0
0 one 1 0
1 0 0 1
ii)
? ? ? ?
?
?
? ? ? ?
?
?
1 one 0 1
0 one 1 0
0 one 1 0
1 0 0 1
*
? ? ? ?
?
?
? ? ? ?
?
?
1 one 0 1
0 one 1 0
0 one 1 0
1 0 0 1
20) discuss the procedure for obtaining a maximum likelihood decoding
function associated with a group code e : Bm ? Bn.
21) discuss the decoding procedure for a group code provided by a parity
check matrix.
22) Let H =
? ? ? ? ? ? ? ?
?
?
? ? ? ? ? ? ? ?
?
?
0 0 1
0 one 0
1 0 0
0 one 1
1 one 0
1 0 0
be a parity check matrix. Decode 011001 relative
to a maximum likelihood decoding function associate with eH.
23) Let H =
? ? ? ? ? ? ? ?
?
?
? ? ? ? ? ? ? ?
?
?
0 0 1
0 one 0
1 0 0
0 one 1
1 one 0
1 0 0
be a parity check matrix. Decode 101011 relative
to a maximum likelihood decoding function associate with eH.

Earning:   Approval pending.