43 research outputs found
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
Quasi-cyclic perfect codes in Doob graphs and special partitions of Galois rings
The Galois ring GR is the residue ring , where
is a basic primitive polynomial of degree over . For any
odd larger than , we construct a partition of GR into -subsets of type and
-subsets of type such that the partition is invariant under
the multiplication by a nonzero element of the Teichmuller set in
GR and, if is not a multiple of , under the action of
the automorphism group of GR.
As a corollary, this implies the existence of quasi-cyclic additive
-perfect codes of index in where is the Doob metric scheme on .Comment: Accepted version; 7 IEEE TIT page