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

Abstract

Let $e \ge 2$ and $r\ge 1$ be integers, and let ${\mathcal {R}}_{e,r}$ denote the Galois ring of characteristic $2^{e}$ and cardinality $2^{e r}.$ The Teichm$\ddot{u}$ller set ${\mathcal {T}}_{r}$ of the Galois ring ${\mathcal {R}}_{e,r}$ can be viewed as the finite field of order $2^r$ under the addition operation $\oplus$ and the multiplication operation of ${\mathcal {R}}_{e,r},$ where for $a,b \in {\mathcal {T}}_{r},$ $a\oplus b$ is the unique element in ${\mathcal {T}}_{r}$ satisfying $a\oplus b = (a+b) ~(\text {mod }2).$ Now a linear code ${\mathscr {C}}$ of length n over ${\mathcal {T}}_{r}$ is said to be k-doubly even if it has a k-dimensional linear subcode ${\mathscr {C}}_{0}$ satisfying ${\textbf {c}}\cdot {\textbf {c}} \equiv 0~(\text {mod 4})$ for all ${\textbf {c}}\in {\mathscr {C}}_0,$ where each ${\textbf {c}}\in {\mathscr {C}}_0$ is viewed as an element of ${\mathcal {R}}_{e,r}^n$ and $\cdot$ denotes the Euclidean bilinear form on ${\mathcal {R}}_{e,r}^n.$ A k-doubly even code of length n and dimension k over ${\mathcal {T}}_{r}$ is simply called a doubly even code. In this paper, we count all doubly even codes over ${\mathcal {T}}_{r}$ 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 ${\mathcal {T}}_{r}.$ We further provide a recursive method to construct self-orthogonal and self-dual codes of the type $\{\texttt {k}_1,\texttt {k}_2,\ldots ,\texttt {k}_e\}$ and length n over ${\mathcal {R}}_{e,r}$ from a $(\texttt {k}_1+\texttt {k}_2+\cdots +\texttt {k}_{\left\lfloor {\frac{e}{2}}\right\rfloor })$-doubly even self-orthogonal code of the same length n and dimension over ${\mathcal {T}}_{r},$ where n is a positive integer and $\texttt {k}_1,\texttt {k}_2, \ldots ,\texttt {k}_e$ are non-negative integers satisfying $2\texttt {k}_1+2\texttt {k}_2+\cdots +2\texttt {k}_{e-i+1} +\texttt {k}_{e-i+2}+\texttt {k}_{e-i+3}+\cdots +\texttt {k}_i \le n$ for , (here $\left\lfloor {\cdot }\right\rfloor$ denotes the floor function and denotes the ceiling function). With the help of this recursive construction method and the enumeration formulae for doubly even codes over ${\mathcal {T}}_{r}$ and their two special classes, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over ${\mathcal {R}}_{e,r}.$ Using these enumeration formulae, we classify all self-orthogonal and self-dual codes of lengths 2, 3 and 4 over ${\mathcal {R}}_{2,2}$ up to monomial equivalence.