課程名稱︰錯誤更正碼
課程性質︰選修
課程教師︰林茂昭
開課學院：電資學院
開課系所︰電機所 電信所
考試日期（年月日）︰2011/11/16
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
The Midterm Exam of Error-Correcting Codes
November 16,2011
1.
(a) Please give a generator matrix of the RM(2,3) Reed-Muller code. (8%)
(b) Please give a generator matrix of the RM(2,4) code in terms of the
generatoer matrices of Reed Muller codes of length 8.(7%)
2.
Consider a cyclic code C of length n that consists of both odd-weight and
even-weight codewords.Let g(X) and A(z) be the generator polynomial and weight
enumerator for this code.Show that the cyclic code generated by (X+1)g(X) has
weight enumerator A1(z)=(1/2)[A(z)+A(-z)]. (10%)
3.
Let ν(X) be the code polynomial in a cyclic doe of length n.Letιbe the
smallest integer such that ν^(ι)(X)=ν(X).
Show that if ι≠0,ι is a factor of n.(10%)
4.
Let α be a primitive element of GF(2^6).
(a) Please show the roots of the generator polynomial of the
double-error-correcting primitive BCH code C1 of length 63.(5%)
(b) Please show the roots of the generator polynomial of the
four-error-correcting primitive BCH code C2 of length 63.(5%)
(c) Please show the roots of the generator polynomial of the dual code of C2.
(5%)
5. Let α be a primitive element of GF(2^8).
(a) Please find all the 17th root of unity in GF(2^8).(5%)
(b) Let β be a primitive 17th root of unity.Please find the BCH bound of a
binary cyclic code of length 17 for which the generator polynomial contains
1,β abd their conjugates as roots.(5%)
(c) Please find the BCH bound of a cyclic code of length 17 over GF(4) for
which the generator polynomial contain β^6,β^8 and their conjugates
as roots.(5%)
6.
(a) Describe and prove the Hamming bound for the (n,k) binary linear code.(8%)
(b) Describe and prove the Singleton bound for the (n,k) linear code.(7%)
7.
Please describe the procedure of syndrome decoding for the (n,k) binary linear
code.(5%)
8.
Please describe the procedure of syndrome encoding for the (n,k) cyclic code.
(5%)
9.
Let α∈GF(q) be an element of order n.The code C of length n is defined by
the set of polynomial A(X) of degree less than k over GF(q).The codeword
╴
a =(a0,a1,…,an-1) are generated by ai=A(α^i),i.e.,
╴
C={a |ai=A(α^i),i=0,1,…,n-1,deg(A(X)) ＜k}.Please show that C is a cyclic
code wuth minimum distance of n-k+1.(10%)
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