POLYNOMIAL CODES AND FINITE GEOMETRIES
Edward F. Assmus · 2003
Contents 1 Introduction 2 2 Projective and a#ne geometries 3 2.1 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A#ne geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Designs from geometries . . . . . . . . . . . . . . . . . . . . . 10 2.4 Codes from designs . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The Reed-Muller codes 12 3.2 Geometries and Reed-Muller codes . . . . . . . . . . . . . . . 16 3.3 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 The group-algebra approach 25 4.1 Elementary results and Berman's theorem . . . . . . . . . . . 26 4.2 Isometries of the group algebra . . . . . . . . . . . . . . . . . 28 4.3 Translation-invariant extended cyclic codes . . . . . . . . . . 30 4.4 The generator polynomials of punctured Reed-Muller codes and their p-ary analogues . . . . . . . . . . . . . . . . . . . . 33 4.5 Orthogonals and annihilators . . . . . . . . . . . . . . . . . . 36 4.6 The codes of the designs