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 + L + 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: k2 s Can detect 2s errors Can correct s errors Generally can correct α erasures and β errors if α + 2 β≤ 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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2 15-853 Page5 A Systematic Code Systematic polynomial-based code p(x) = a k-1 x k-1 + L + a 1 x + a 0 Message : (a k-1 , …, a 1 , a 0 ) Codeword : (a k-1 , …, a 1 , a 0 , p(1), p(2), …, p(2s)) This has the advantage that if we know there are no errors, it is trivial to decode. The version of RS used in practice uses something slightly different than p(1), p(2), … This will allow us to use the “ Parity Check ”ideas from linear codes (i.e., Hc T = 0?) to quickly test for errors. 15-853 Page6 Reed-Solomon Codes in the Real World (204,188,17) 256 : ITU J.83(A) 2 (128,122,7) 256 : ITU J.83(B) (255,223,33) 256 : Common in Practice – Note that they are all byte based (i.e., symbols are from GF(2 8 )). Decoding rate on 1.8GHz Pentium 4: – (255,251) = 89Mbps – (255,223) = 18Mbps Dozens of companies sell hardware cores that operate 10x faster (or more) – (204,188) = 320Mbps (Altera decoder) 15-853 Page7 Applications of Reed-Solomon Codes Storage : CDs, DVDs, “hard drives”, Wireless : Cell phones, wireless links Sateline and Space : TV, Mars rover, … Digital Television : DVD, MPEG2 layover High Speed Modems : ADSL, DSL, . . Good at handling burst errors. Other codes are better for random errors. – e.g., Gallager codes, Turbo codes 15-853 Page8 RS and “burst” errors They can both correct 1 error, but not 2 random errors.
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ecc2 - Viewing Messages as Polynomials 15-853:Algorithms in...

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