Construction and enumeration of self-orthogonal and self-dual codes over Galois rings of even characteristic

Let e >= 2 and r >= 1 be integers, and let R-e,R-r denote the Galois ring of characteristic 2(e) and cardinality 2(er). The Teichmuller set T-r of the Galois ring R-e,R-r can be viewed as the finite field of order 2(r) under the addition operation. and the multiplication operation of R-e,R-r, where for a, b is an element of Tr, a circle plus b is the unique element in T-r satisfying a circle plus b = (a + b) (mod 2). Now a linear code C of length n over T-r is said to be k-doubly even if it has a k-dimensional linear subcode C-0 satisfying c center dot c = 0 (mod 4) for all c is an element of C-0, where each c is an element of C-0 is viewed as an element of R-e,r(n) and center dot denotes the Euclidean bilinear form on R-e,r(n). A k-doubly even code of length n and dimension k over T-r is simply called a doubly even code. In this paper, we count all doubly even codes over T-r and their two special classes, viz. the codes containing the all-one vector and the codes that do not contain the all-one vector by studying the geometry of a certain special quadratic space over T-r. We further provide a recursive method to construct self-orthogonal and self-dual codes of the type {k(1), k(2),..., k(e)} and length n over R-e,R-r from a (k(1)+ k(2)+ ...+ k(left) (perpendiculare/2right perpendicular))-doubly even self-orthogonal code of the same length n and dimension k(1) + k(2) + ... + k(inverted right perpendiculare/2inverted left perpendicular) over T-r, where n is a positive integer and k(1), k(2),..., k(e) are non-negative integers satisfying 2k(1) + 2k(2) + ... + 2k(e-i+1) + k(e-i+2) + k(e-i+3) + ... + k(i) <= n for inverted right perpendiculare+1/2inverted left perpendicular <= i <= e, (here left perpendicuar center dot right perpendicuar denotes the floor function and inverted right perpendicular center dot inverted left perpendicular denotes the ceiling function). With the help of this recursive construction method and the enumeration formulae for doubly even codes over T-r and their two special classes, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over R-e,R-r. Using these enumeration formulae, we classify all self-orthogonal and self-dual codes of lengths 2,3 and 4 over R-2,R-2 up to monomial equivalence.