作者suhorng ( )
看板NTU-Exam
标题[试题] 100下 黄汉水 编码学期末考
时间Thu Jun 14 19:53:19 2012
课程名称︰编码学
课程性质︰选修
课程教师︰黄汉水
开课学院:
开课系所︰数学系
考试日期(年月日)︰2012/06/14
考试时限(分钟):170分钟
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
一 The following ISBN have been received with smudges.
What are the missing digits? (10%)
0-75a-57297-1, 1-23-52b2489
二 Let C be a binary linear code having the following parity-check matrix
┌ ┐
│ 1 1 0 1 0 0 0 │
│ │
│ 1 0 1 1 1 0 0 │
H = │ │ (30%)
│ 0 0 1 1 0 1 0 │
│ │
│ 0 0 1 0 1 1 1 │
└ ┘
(1) Find a generator matrix for C in standard form.
(2) Find a parity-check matrix for C in standard form.
(3) Find the minimum distance of C.
(4) Decode the received vectors 1011100, 0111101, 0110011.
(5) Is C a cyclic code? Prove your answer.
(注:(4)老师後来特别说若有minimum distance相同的codeword, 任一即可)
三 Let C be a ternary linear code having the following generator matrix
┌ ┐
│ 1 2 0 1 │
G = │ │ (30%)
│ 2 2 2 0 │
└ ┘
(1) Find a parity-check matrix for C in standard form.
(2) Find a generator matrix for C in standard form.
(3) Find the minimum distance of C.
(4) Decode the received vectors 2202, 1100.
(5) Is C a cyclic code? Prove your answer.
四 Let F_5 = {0,1,2,3,4} be the finite field and
T_k = { f(x)∈F_5[x] | f(x) is a monic irreducible polynomial of degree k}
(1) Find the set T_1, T_2. (7%)
(2) How many polynomials in the set T_3, T_4? Prove your answer. (8%)
五 Let C be a [6, 4, 3] linear code over the finite field F_5 = {0,1,2,3,4}.
How many codewords in C of weight 3? Prove your answer. (15%)
[Hint: Is C a perfect code?]
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