Unformatted text preview: (e) Assuming the string 010101 is received, and at most one error occurred (if any), determine the codeword that was sent. 3. Suppose that G is a ﬁnite group of order < 100 and that H and K are subgroups of G of orders 18 and 30, respectively. (a) What is the order of G ? Why? (b) How many left cosets of H are there? Explain. 4. (a) Let f 1 and f 2 be elements of F [ x ], where F is some ﬁeld. Suppose g = gcd( f 1 ,f 2 ) in F [ x ]. Let V := { α ∈ F  f 1 ( α ) = f 2 ( α ) = 0 } be the set of common roots of f 1 and f 2 in F , and U := { α ∈ F  g ( α ) = 0 } be the set of roots of g in F . Show V = U . (b) Find gcd( x 4 + 3 x 3 + 2 x + 1 ,x 3 + x 2 + x + 1). (c) Use parts (a) and (b) to ﬁnd all common rational roots of the polynomials x 4 + x 3 + x + 1 and x 3 + 2 x 2 + 2 x + 1 . 1...
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 Spring '08
 BILLERA
 Math, Algebra, Cryptography, Hamming Code, parity check matrix, common rational roots

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