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By Scott A. Vanstone
Five. 2 earrings and beliefs 148 five. three beliefs and Cyclic Subspaces 152 five. four Generator Matrices and Parity-Check Matrices 159 five. five Encoding Cyclic Codest 163 five. 6 Syndromes and straightforward deciphering systems 168 five. 7 Burst mistakes Correcting a hundred seventy five five. eight Finite Fields and Factoring xn-l over GF(q) 181 five. nine one other strategy for Factoring xn-l over GF(q)t 187 five. 10 workouts 193 bankruptcy 6 BCH Codes and limits for Cyclic Codes 6. 1 creation 201 6. 2 BCH Codes and the BCH sure 205 6. three Bounds for Cyclic Codest 210 6. four interpreting BCH Codes 215 6. five Linearized Polynomials and discovering Roots of Polynomialst 224 6. 6 workouts 231 bankruptcy 7 blunders Correction suggestions and electronic Audio Recording 7. 1 creation 237 7. 2 Reed-Solomon Codes 237 7. three Channel Erasures 240 7. four BCH deciphering with Erasures 244 7. five Interleaving 250 7. 6 blunders Correction and electronic Audio Recording 256 7.
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Additional info for An introduction to error correcting codes with applications
A second inner-sphere of radius Rin touches the mid-point of each face, whilst a third mid-sphere of radius Rint touches the mid-point of each edge. These radii, together with the surface area A, volume V and the dihedral angle between any neighboring pair of faces ᭝ can be expressed in terms of the parameters p, q, f and s, where p ϭ the number of edges in a face q ϭ the number of edges associated with a vertex f ϭ the number of faces s ϭ the edge length The following steps show the formulas used for calculating Rin, Rint, Rc, ⌬, A and V.
1 Internal and external angles of a polygon The internal angles of an n-gon sum to (n Ϫ 2) ϫ 180°. Quadrilateral (n ؍4) a4 4 ∑ ai ϭ 360Њ a3 a1 a2 iϭ1 The external angles of an n-gon sum to 360°. 2 Alternate internal angles of a cyclic polygon The alternate internal angles of a cyclic n-gon sum to (n Ϫ 2) ϫ 90° [n у 4 and is even]. 3 Area of a regular polygon Area of a polygon using the number of edges Area ϭ ⎛p⎞ 1 2 ns cot ⎜ ⎟ 4 ⎝n⎠ or ⎛p⎞ ⎛p⎞ Area ϭ nr 2 sin ⎜ ⎟ cos ⎜ ⎟ ⎝n⎠ ⎝n⎠ or ⎛ 2p ⎞ 1 Area ϭ nr 2 sin ⎜ ⎟ 2 ⎝ n ⎠ 20 where Geometry for computer graphics n ϭ number of sides s ϭ length of side r ϭ radius of circumscribed circle Area of a polygon using Cartesian coordinates Area ϭ 1 n −1 Ϫ yi xiϩ1(mod n) ) ∑ (x y 2 iϭ0 i iϩ1(mod n) where the n vertices (x, y) are deﬁned in counter-clockwise sequence.
A c ϭ aЈ cЈ a ϭ aЈ Second triangle bЈ aЈ cЈ c aЈ a a aЈ 12 Geometry for computer graphics Two corresponding angles are equal. 3 Congruent triangles Two triangles are congruent (identical ഡ) if corresponding sides and angles are equal. Conditions for congruency First triangle Second triangle Three sides are equal. a ϭ aЈ b ϭ bЈ c ϭ cЈ c b cЈ bЈ a aЈ Two sides and the included angle are equal. a ϭ aЈ b ϭ bЈ a ϭ aЈ b bЈ a aЈ a aЈ One side and the adjoining angles are equal. 4 Theorem of Pythagoras Pythagorean formula In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
An introduction to error correcting codes with applications by Scott A. Vanstone