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

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

(New Syllabus w.e.f. June 2008)
Class : S.Y. B. Sc. Subject : Mathematics
Paper : MTH – 212 (B) (Computational Algebra)
(Calculus of Several Variables)

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1 : ques. of two marks
1) describe i) a message ii) a word
2) describe i) an (m , n) encoding function ii) an alphabet
3) describe i) a code word ii) a code
4) describe weight of a word. obtain the weight of a word 110110101.
5) describe parity check code. If e : B4 ? B5 is a parity check code then
obtain e(1010) and e(1011).
6) describe the Hamming distance ranging from the words x, y ? Bm. If e : B4
? B5 is a parity check code then obtain d(e(0110) , e(1101))
7) If e : B4 ? B5 is a parity check code then obtain
i) d(e(1011) , e(1101)) ii) d(e(0011) , e(1001))
2 3
1
4
5
6
10
8) describe the minimum distance of an encoding function. If e : B2 ? B4
is encoding function described by e(b1b2) = b1b2b1b2 then obtain minimum
distance of e.
9) obtain the minimum distance of (2 , 3) parity check code.
10) If e : B2 ? B4 is encoding function described by e(b1b2) = b1b2b2b1b2,
then obtain minimum distance of e.
11) describe Parity check matrix. If H =
? ? ?
?
?
? ? ?
?
?
0 1
1 0
1 1
is a parity check matrix
then obtain (1,3) group code eH : B1 ? B3.
12) describe the minimum distance of a decoding function.
13) obtain weight of every of the subsequent words in B4 : x = 1010 , y =
1110 , z = 0000 , w = 1111. Also obtain d(x , y) , d(z , w).
14) obtain weight of every of the subsequent words in B7 : x = 1100010, y =
1010110, z = 1111111, w = 1110101. Also obtain d(x , y) , d(z , w).
15) calculate i) ??
?
??
?
0 one 1
1 one 0 ? ??
?
??
?
0 one 1
1 one 1 ii)
? ? ?
?
?
? ? ?
?
?
1 0 1
0 one 1
1 0 1
*
? ? ?
?
?
? ? ?
?
?
1 0 1
0 one 1
1 one 0
16) calculate i)
? ? ?
?
?
? ? ?
?
?
1 0 1
0 one 1
1 0 1
?
? ? ?
?
?
? ? ?
?
?
1 0 1
0 one 1
1 0 1
ii)
? ? ?
?
?
? ? ?
?
?
0 1
1 0
1 1
* ??
?
??
?
1 one 0
0 one 1
17) If Bm = B×B× - - - ×B (m factors) is a group under the binary
operation ? then i) obtain the identity element of Bm.
ii) obtain inverse of x ? Bm. iii) Write the order of Bm.

Earning:   Approval pending.