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# ecc2 - Viewing Messages as Polynomials 15-853:Algorithms in...

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1 15-853 Page1 15-853:Algorithms in the Real World Error Correcting Codes II – Cyclic Codes – Reed-Solomon Codes 15-853 Page2 Viewing Messages as Polynomials A (n, k, n-k+1) code: Consider the polynomial of degree k-1 p(x) = a k-1 x k-1 + ± + a 1 x + a 0 Message : (a k-1 , …, a 1 , a 0 ) Codeword : (p(1), p(2), …, p(n)) To keep the p(i) fixed size, we use a i ± GF(p r ) To make the i distinct, n < p r Unisolvence Theorem : Any subset of size k of (p(1), p(2), …, p(n)) is enough to (uniquely) reconstruct p(x) using polynomial interpolation, e.g., LaGrange’s Formula. 15-853 Page3 Polynomial-Based Code A (n, k, 2s +1) code: k 2s Can detect 2s errors Can correct s errors Generally can correct ² erasures and ³ errors if ² + 2 ³ ´ 2s n 15-853 Page4 Correcting Errors Correcting s errors : 1. Find k + s symbols that agree on a polynomial p(x). These must exist since originally k + 2s symbols agreed and only s are in error 2. There are no k + s symbols that agree on the wrong polynomial p’(x) - Any subset of k symbols will define p’(x) - Since at most s out of the k+s symbols are in error, p’(x) = p(x)

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ecc2 - Viewing Messages as Polynomials 15-853:Algorithms in...

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