61 research outputs found
Erasure Techniques in MRD codes
This book is organized into six chapters. The first chapter introduces the
basic algebraic structures essential to make this book a self contained one.
Algebraic linear codes and their basic properties are discussed in chapter two.
In chapter three the authors study the basic properties of erasure decoding in
maximum rank distance codes. Some decoding techniques about MRD codes are
described and discussed in chapter four of this book. Rank distance codes with
complementary duals and MRD codes with complementary duals are introduced and
their applications are discussed. Chapter five introduces the notion of integer
rank distance codes. The final chapter introduces some concatenation
techniques.Comment: 162 pages; Published by Zip publishing in 201
On Quadratic Residue Codes Over Finite Commutative Chain Rings
Codes over finite rings were initiated in the early 1970s, And they have received much attention after it was proved that important families of binary non-linear codes are images under a Gray map of linear codes over Z4. In this thesis we consider a special families of cyclic codes namely Quadratic residue codes over finite chain rings F2 + uF2 with u2 = 0 and F2 + uF2 + u2F2 with u3 = 0. We study these codes in term of their idempotent generators, and show that these codes have many good properties which are analogous in many respect to properties of Quadratic residue codes over finite fields, also, we study Quadratic residue codes over the ring Z2m, and then generalize this study to Quadratic residue codes over finite commutative chainring Rm-1 = F2 + uF2 + : : : + um-1F2 with um =
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
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