By Vera Pless(auth.)
Chapter 1 Introductory options (pages 1–15):
Chapter 2 priceless heritage (pages 17–38):
Chapter three A Double?Error?Correcting BCH Code and a Finite box of sixteen parts (pages 39–49):
Chapter four Finite Fields (pages 51–66):
Chapter five Cyclic Codes (pages 67–83):
Chapter 6 workforce of a Code and Quadratic Residue (QR) Codes (pages 85–108):
Chapter 7 Bose?Chaudhuri?Hocquenghem (BCH) Codes (pages 109–222):
Chapter eight Weight Distributions (pages 123–141):
Chapter nine Designs and video games (pages 143–167):
Chapter 10 a few Codes Are distinctive (pages 169–188):
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Extra info for Introduction to the Theory of Error-Correcting Codes, Third Edition
The analogous theorem is true for monic irreducible polynomials. The proof of this theorem is Problem 9. Theorem 24. Every monic polynomial over a field F can be expressed uniquely as a product of irreducible monic polynomials over F. If fix) is a polynomial over a field F, a e F, and fia) = 0, then a is called a root of fix). Theorem 25. If fix) is a polynomial over a field F, then a is a root of fix) iff fix) - gixXx - a). Proof. Clearly, if fix) = gixXx - a), then fia) = 0. For any a, by the division algorithm, we can write fix) = ix - a)bix) + rix) where the degree of rix) is less than 1.
6. What can you say about the coset weight distribution of a [20,12] double-error-correcting binary code? 7. Show that a binary code can correct all single errors iff any parity check matrix has distinct nonzero columns. 8. The code Ham(4,2) has the following parity check matrix H: H= 0 0 0 u 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 ij Decode the vectors yx = (1,1,1,0,1,1,1,0,0,0,0,0,0,0,0) and y2 = (1,1,0,0,1,1,0,0,1,1,0,0,1,1,1) for Ham(4,2).
Theorem 14. If u is a vector in C of weight s, then there is a dependence relation among s columns of any parity check matrix of C, and conversely, any dependence relation among s columns of a parity check matrix of C yields a vector of weight s in C. Proof Let H be a parity check matrix of C. Since u is orthogonal to each row of H, the s components in u that are nonzero are the coefficients of the dependence relation of the s columns of H corresponding to the s nonzero components. The converse holds by the same reasoning.