Research On Code Over Ring And Its Property

In recent years, many researchers studing coding theories have shifted their eyes from those of the finite field to those of the finite ring, especially , the research on Z₄ code. These researchers try to integrate Z₄ code with binary code therough Gray mapping. Based on the research achieved by forerunners, the author proposes his own perspectives upon this topic as follows, whichcomposes of the core of this thesis, property.1 The author introducts an isometry φ^k for k≥1 between codes over Z_p^k+1, and codes over Z_p² is introduced and is used to give a generalization ofGray map φ: Z₄ⁿ →F₂²n .Furthermore, by means of this isometry, the concept of negacyclic codes is extended to codes over the ring Z ₍p^k+1 .A characterization ofthis codes in terms of their image under φ^k is given. It is also shown that the generalized Gray map image of (1- p^k)- cyclic is a distance invariant (not necessary linear) quasi-cyclic code. Finally, some linear (l - p^k')- cyclic codes are discussed.2 The property of isometric mapping of φ^k :Z_p^k+1n→Z_p2^pk+1n is discussed according to the author's own understanding.3 (l + u)-cyclic code of even length over the F₂+uF₂ ring is studied and further classified.