MATH3411 INFORMATION, CODES & CIPHERS Test 1, Session 2 2011, SOLUTIONS
Multiple choice: c,b,c,d,a True/False: T, F, F, T, T
(b): third block has 3 ones and the 20th bit is in that block. (c): The syndrome is 010.
(a): create a tree or 14 + 81 = (0.011)2.
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(b) False: Sphere packing bound implies |C| = 2k ≤ 2n
6. (a)True.w=d=7=2×3+1sothreecanbecorrected.
1+n+ 12n(n−1) for 2-error correcting and n = 8, k = 3 does not satisfy this in-
(c) False: 1010 = c3+c3 = c4+c1. Alternatively, the Kraft-McMillan constant for this code is 98, so by the Kraft-McMillan theorem no UD code with the parameters of this code exists.
(d) True: K = 1 + 4 × 1 + 2 × 1 < 1. 3 9 27
(e) True: Mp=pandsumofentriesofpis1.
In order the codewords are 00, 10, 010, 110, 111, 0110, 0111. (b) Average length is 51.
Multiple Choice: d, d, b, c, d True/False: F, T, T, F, T
(d): last block has 3 ones and the 35th bit is in that block
(b): Sphere packing bound implies |C| = 2 ≤ 1 + n + 21n(n − 1) for
2-error correcting. The right-hand side of this inequality is 29 ≈ 4.4,
(c): Since that codeword has weight 7, w = d = 7 = 2×3+1 so three
can be corrected.
(d): The Kraft-Macmillan constant is clearly greater than 1 in the first
twocasesandforr=4weget3×1 +4× 1 + 1 >1,butforr=5,
K=3×1+4×1+1 <1. 5 25125
(a) False: second check equation is not satisfied. (b) True: multiply out the matrix and vector.
(c) True: it’s an I-code (draw a tree).
(d) False: c5 = 101 either by the tree or 41 +3× 81 = 12 + 81 = (0.101)2.
(e) True: Mp=pandsumofentriesofpis1.
(a) In order the codewords are 01, 11, 000, 001, 101, 1000, 1001 (b) Average length is 52 = 13.
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