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## Sp Top Probability

by: Desmond Cormier

28

0

6

# Sp Top Probability MATH 350

Desmond Cormier

GPA 3.9

Julie Clark

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COURSE
PROF.
Julie Clark
TYPE
Class Notes
PAGES
6
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 6 page Class Notes was uploaded by Desmond Cormier on Monday October 12, 2015. The Class Notes belongs to MATH 350 at Hollins University taught by Julie Clark in Fall. Since its upload, it has received 28 views. For similar materials see /class/222133/math-350-hollins-university in Mathematics (M) at Hollins University.

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Date Created: 10/12/15
Math 350 Codes amp Ciphers March 3 2009 Adding Vectors mod 2 gt restart gtwithlinalg We can define row vectors using transpose gtv1transposeltlt1010101gtgt v2 transpose ltlt0 1 11 0 0 0gtgt v11010101 v201110 0 0 gtvmap modpmataddv1v2 2 v1101101 Or we can define the vectors as matrices gtv3matrix171010101 v4matrix171111101 v31010101 v41111101 gtvmap modpmataddv3v4 2 v0101000 Generator Matrix for the Hamming 74 code 10000 1 G1 1100 H0 1010 1 1001 1 1 0 000 Math 350 Codes amp Ciphers March 3 2009 Using Maple t0 Encode with the Hamming7 4C0de gt restart gt with linalg with LinearAlgebra gtGmatrix47 11100001001100010101 IOIlIlIOIlIOIOIl 1 1 1 0 0 0 0 G1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 gt a matrixl4 Oll0 a0 1 1 0 gt b mapmodpmultiplyaG 2 b1100110 Now suppose we want to add an 8th parity check digit gt I7 IdentityMatrix 7 gtC1 matrix711111111 1 1 1 C1 1 1 1 1 gtCMconcatI7C1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 CM0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 gt cw map modp multiply b CM 2 cw11001100 March 3 2009 Math 350 Codes amp Ciphers The Family of Hamming Codes mu 3 3 3 3 3 f 4 6 n 8 8a m as 2 6 4 s x r 1 0 0 4 0 2 1 w 7 4 6 1 n 8 k B H 3 9 4 1 5 7 0 R 9 1 m 1 6 7 m2 1 4 1 2 5 k 1 m 5 1 3 3 7 1 3 6 n m 2 3 4 5 6 Generator Matrix for Hamming1511 code 000000000000 1 1 1 1 1 1 1 10000000000 0 0 0 000000000 1 1 00 00000000 1 0 000000 000000 1 10100000 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 G 10000 100 000 1 1 1000 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 10000 0 0 100000 0 0 1000000 parity check matrix for Hamming74 code 1010101 0110011 0001111 Math 350 Codes amp Ciphers March 3 2009 ErrorCorrecting with the Hamming84 Code gt restart gtwithlinalg withLinearAlgebra gt Ptransposeltlt001gtlt010gtlt011gtlt100gtlt101gtlt110gtlt1 1 1gtgt I I D D D D OOO D D OOD D IO t IOHOt IOH And now we input a length 8 received vector that we want to decode gtrvmatrix1810101000 rv10101000 gtparitymapmodpsumrv1j j1 8 2 parity 1 gt chckword map modp multiply delcols rv 8 8 P 2 chckword 1 1 gtif parity 1 then one error in word or 8th digit We compute the location of the error if any and correct the error in our word gt if 11 12 13 gt 0 then gt digiterror chckword1 3 2chckword1 2 4chckword1 1 gt rv1digiterror rv1digiterror1mod 2 gt printquotsingle Error Corrected in positionquot digiterror gt u1rv13u2rv15u3rv16u4rv17 gt printut t14 gt else gt printquotError in Check Digit Onlyquot gt u1 rv13 u2 rv15 u3 rv16 u4 rv17 gt printut t1 4 gt end if 4 Math 350 Codes amp Ciphers March 3 2009 gtelif 11 12 13 0 then gt printquotNo errorquot gt u1 rv13 u2 rv15 u3 rv16 u4 rv17 gt printut t1 4 gt else printquotDouble Error Detectedquot gtend if quotSingle Error Corrected in positionquot 7 1101 Math 350 Codes amp Ciphers March 3 2009 Homework Exercises 4 Due 31009 1 Without using Maple locate the position of any errors and then correct them in the following received words from the Hamming74 code a r1 0101111 b r2 0001101 c r3 0100011 d r4 0110011 2 Let C be a binary linear code Prove that the code Cobtained by adding an overall parity check digit to C is linear closed under the operation of addition 3 For each of the generator matrices below list all the codewords in the code and find the minimum distance of the code 1 1 1 1 0 1 0 0 1 1 0 aG1 bG2010101 0 0 1 1 1 0 0 1 0 1 1 4 Use Maple to help you fill in the table below Information H 74 H 84 parity Information H 74 H 8l4 parity Digits codeword check digit Digits codeword check digit 0000 1001 0001 1010 0010 0101 0100 0111 1000 1011 0011 1101 0110 1110 1100 1111

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