[試題] 100下 黃漢水 編碼學期末考
課程名稱︰編碼學
課程性質︰選修
課程教師︰黃漢水
開課學院：
開課系所︰數學系
考試日期（年月日）︰2012/06/14
考試時限（分鐘）：170分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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.
(註:(4)老師後來特別說若有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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