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Codes over the chain ring are obtained by writing special matrices. Gray images of these codes are binary codes. It is shown that first repeat code, second repeat code, even code and odd code are either equivalent or equal to these codes. The definitions of direct sum and direct product of these codes were given. Moreover dual codes were classed. Self dual codes and self orthogonal codes were established.

In this study the residual algebraic free extension of a valuation on a field K to K(x) is studied. It is assumed that v is a valuation on K with rankv = 2 and the residual algebraic free extension w of v to K(x) with rankw = 3 is defined for a special case.

This paper deals with constacyclic codes over a new ring. Gray images of these codes over those rings are obtained. The relations between these codes and their Gray images are given.

In this study, certain matrices are obtained using the elements of a finite chain ring. Then using thesematrices as generator matrices; certain codes and their duals are obtained. Moreover relations betweenthese codes, binary codes and Hadamard codes are explained.

In this study, some special matrices are constructed by choosing certain elements from finite rings and certain codes are written by using these matrices. These codes are extended to a field. Moreover these codes are classified and more good codes are written.

Let υ be a valuation on K with value group G υ , residue field k υ , rank υ = t and K ( x 1 , …, x n ) be the field of rational functions over K with n variables. If G is the direct sum of G 1 and d infinite cyclic groups where G 1 is a totally ordered group containing G υ as an ordered subgroup with [ G 1 : G υ ] k ′ is a finite field extension of k υ then there exists a residual transcendental extension u of υ to K ( x 1 , …, x n ) such that rank u = t + d , G u = G the algebraic closure of k υ in k υ is k ′ and trans deg k u / k υ = n − d .

Let v be a valuation of a field K , G v its value group and k v its residue field. Let w be an extension of v to K ( x 1 , … , x n ). w is called a residual transcendental extension of v if k w / k v is a transcendental extension. In this study a residual transcendental extension w of v to K ( x 1 , … , x n ) such that transdegk w / k v = n is defined and some considerations related with this valuation are given.

Let v be a valuation of a field K with a value group and a residue field , w be an extension of v to . Then w is called a residual algebraic torsion extension of v to if is an algebraic extension and is a torsion group. In this paper, a residual algebraic torsion extension of v to is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to with given residue field and value group is studied. MSC:12J10, 12J20, 12F20.