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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