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

4) If minimum distance of an encoding function e : Bm ? Bn is k
then e can detect - - - -
a) k or fewer errors b) less than k errors
c) more than k errors d) k + one errors
5) An encoding function e : Bm ? Bn is a group code if
a) Ran{e} is a subgroup of Bm. b) Ran{e} is a subgroup of Bn.
c) Ran{e} is not a subgroup of Bm. d) none of these
6) If d : Bn ? Bm is a (n,m) decoding function then - - - -
a) m = n and d is onto b) m = n and d is 1 one
c) m = n and d is onto d) m = n and d is 1 one
7 Let e : Bm ? Bn be an encoding function with minimum distance
2k + 1. If d is maximum likehood decoding function associated
with e then [ed] can accurate - - - -
a) more than k errors b) more than 2k + one errors
c) k errors d) less than or equal to k errors
8) If B = {0 , 1} then order of a group B4 = - - - -
a) two b) four c) eight d) 16
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3 : ques. of three marks
1) Let x, y be elements of Bm. Show that i) d(x , y) = 0
ii) d(x , y) = 0 ? x = y.
2) Let x, y, z be elements of Bm. Show that i) d(x , y) = d(y , x)
ii) d(x , y) = d(x , z) + d(z , y).
1) If minimum distance of an encoding function e : Bm ? Bn is at lowest
k + one then prove that e can detect k or fewer errors.
2) If an encoding function e : Bm ? Bn can detect k or fewer errors then
prove that its minimum distance is at lowest k + 1.
3) Let e : Bm ? Bn be a group code. Prove that the minimum distance of
e is the minimum weight of a non zero code.
4) Let m < n, n – m = r and x = b1b2 - - - - bmx1x2 - - - xr ? Bn and x * H
= 0 , where H is the parity check matrix of order nxr. Show that there
exists an encoding function eH : Bm ? Bn such that x = eH(b), for a few
b ? Bm.
5) Consider (3 , 6) encoding function e : B3 ? B6 described by e(000) =
000000, e(001) = 001100, e(010) = 010011, e(100) = 100101, e(011)
= 011111, e(101) = 101001, e(110) = 110110, e(111) = 111010. Show
that e is a group code.
6) Consider (3 , 6) encoding function e : B3 ? B6 described by e(000) =
000000, e(001) = 001100, e(010) = 010011, e(100) = 100101, e(011)
= 011111, e(101) = 101001, e(110) = 110110, e(111) = 111010. How

Earning:   Approval pending.