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hw4_810_ss11

# hw4_810_ss11 - r j Remark The calculations required here...

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Math 810, Spring 2011 – HW 4 Due Monday, 28 March 2009 1. Consider the ternary [13 , 10] Hamming code with check matrix 1 0 1 2 0 1 2 0 1 2 0 1 2 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 1 2 2 2 . Decode the received word (1 , 0 , 1 , 1 , 2 , 1 , 2 , 2 , 2 , 2 , 2 , 2 , 2) . 2. Consider the code C = GRS 6 , 2 ( α , v ) over the ﬁeld F 11 of the integers modulo 11, where α = (1 , 2 , 4 , 8 , 5 , 10) and v = (1 , 1 , 1 , 1 , 1 , 1). Suppose a codeword has been transmitted and we receive the word r = (4 , 3 , 2 , 1 , 7 , 10) . If there were no errors, then there would be a single polynomial f ( x ) = a + bx F 11 [ x ] 2 with r = ev α,v ( f ( x )). That is, for each pair of positions i,j , it would be true that f ( α i ) = a + i = r i and f ( α j ) = a + j = r j . But this is not the case. ( a ) For each of the 15 distinct pairs of positions i,j with i < j , ﬁnd constants a i,j and b i,j in F 11 so that the linear polynomial f i,j ( x ) = a i,j + b i,j x has f i,j ( α i ) = r i and f i,j ( α j ) =
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Unformatted text preview: r j . [ Remark. The calculations required here are linear interpolations—ﬁnding the equation of the line through two given points ( α i ,r i ) and ( α j ,r j ).] ( b ) In ( a ), one polynomial f ( x ) should have come up as f i,j ( x ) much more often than any of the others. Give f = ev α,v ( f ( x )). We decode r to f = ev α,v ( f ( x )). ( c ) Give the minimum distance of the code C , and use that to justify the decoding guess of ( b ). 3. Problem 5.1.2 from the Notes . ( Hint: Consider canonical generator matrices for the two versions of the code. Remember that permuting the rows of a generator matrix for C gives a second generator matrix for C .)...
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