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MATH32032 Assignment

创建日期：2024-04-12 15:26:56

所属分类：数学mathematics

标签： MATH32032

Assignment

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MATH32032

SECTION A Answer ALL questions in this section (40 marks in total) A1. (a) Define what is meant by: —

a word x of length n in a finite alphabet F ; — a code C of length n in a finite alphabet F ; —

the Hamming distance d(x, y) between two words x, y of length n; — a nearest neighbour of a word x in a code C.

(b) Let C be the binary repetition code of length 6. List the codewords of C. Write down words x, y ∈ F62

such that x has exactly one nearest neighbour in C, and y has more than one nearest neighbour in C. [10 marks] A2.

(a) Explain what is meant by a standard array for a linear [n, k, d]q-code C. State the number of rows and the number

of columns in a standard array for C. Explain how to decode a received vector using the standard array. You are now

given that D = {0000, 0111, v, 1110} is a binary linear code. (b) Prove that the vector v is 1001. Assuming that a codevector of D is sent via BSC(p), show that, with probability p2 − p4, the received vector will contain an error which is not detected. (c) Construct a standard array for D. Use the standard array you have constructed to give an example where one bit error occurs in a codevector c of D and is not corrected by the standard array decoder. (d) Assume that y, z ∈ F42, y 6= z, and that H is a parity check matrix for the code D such that yHT = zHT . Can y and z occur in the same column of the standard array that you constructed in part (c)? Justify your answer briefly. [20 marks] A3. (a) Define what is meant by a line in the vector space Frq. Explain how to construct a check matrix H for a Hamming Ham(r, q)-code. State without proof the number of rows and the number of columns of H. (b) Write down a check matrix of a ternary Ham(2, 3)-code. (c) Write down a check matrix of a Ham(2, 5)-code C ⊆ F65 such that C contains the vector 111111. [10 marks] Page 2 of 3 P.T.O. MATH32032 SECTION B Answer TWO of the three questions in this section (40 marks in total). If more than TWO questions from this section are attempted, then credit will be given for the best TWO answers. B4. In this question, F is a finite alphabet of cardinality q, and C ⊆ F n is a code of cardinality M > 1. (a) Define what is meant by the minimum distance, d(C), of the code C. (b) Prove the Singleton bound which says that M 6 qn−d(C)+1. (c) Let r be an integer between 0 and n. Define the Hamming sphere Sr(u) of radius r centred at u ∈ F n. State and prove the formula for the cardinality of Sr(u) in terms of r, n and q. (d) State without proof the Hamming bound on M in terms of n, q and d(C). Define what is meant by saying that C is a perfect code. (e) Show that if n = q = 60 and C is perfect, then d(C) 6= 7. Hint: you may use the Singleton bound. [20 marks] B5. In this question, C ⊆ Fnq is a linear code of dimension k. (a) Define what is meant by the inner product x · y of two vectors x, y ∈ Fnq , and what is meant by the dual code C⊥. State without proof the length and the dimension of C⊥. (b) Recall that C is self-orthogonal if C ⊆ C⊥. Prove that if C has a generator matrix G such that GGT is a zero matrix, then C is self-orthogonal. (c) A self-orthogonal code D ⊆ F4q has generator matrix [ 1 2 −1 2 2 1 2 −1 ] . Find a prime number q > 2 for which this is possible. Then write down a check matrix for D, explaining how it is obtained. (d) Assume that q = 2 and C ⊆ C⊥. Let WC(x, y) = ∑n i=0Aix n−iyi be the weight enumerator of C. Prove that A2 6 k and that A2(A2 − 1)/2 6 A4. [20 marks] B6. In this question, we assume that vectors in Fnq are identified with polynomials, in Fq[x], of degree less than n in the usual way. (a) Let C ⊆ Fnq be a cyclic code. Explain what is meant by a generator polynomial of C. Show that there exists at most one generator polynomial of C. (b) Let Z ⊆ F87 be the cyclic code with generator polynomial x − 1. Write down a generator matrix for Z. Prove that Z = {(v1, v2, v3, v4, v5, v6, v7, v8) ∈ F87 : ∑8 i=1 vi = 0 in F7}. You may use the factorisation (x− 1)(x+1)(x2+1)(x2− 3x+1)(x2+3x+1) of x8− 1 into irreducible monic polynomials in F7[x] to answer the following questions. (c) Determine: (i) the total number of cyclic codes C ⊆ F87; (ii) the number of cyclic codes C ⊆ F87 such that C ⊆ Z where Z is the code from part (b). Justify your answers briefly. (d) Prove that none of the cyclic codes in F87 is a Hamming code. Any facts from the course can be freely used. [20 marks] END OF EXAMINATION PAPER

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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