This paper investigates the existence, enumeration, and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic p 2 and order p 4 with p an odd prime. When p ≡ 3 (mod 4), we give a method to construct a duality preserving bijective Gray map from such a Galois ring to Z 2 p 2. Closed formed enumeration formulas for double circulant codes that are self-dual (resp. LCD) are derived as a function of the length of these codes. Using random coding, we obtain families of asymptotically good self-dual and LCD codes over Z p 2 with respect to the metric induced by the standard Fp-valued Gray maps.

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The Doob graph D(m, n) is the Cartesian product of m > 0 copies of the Shrikhande graph and n copies of the complete graph of order 4. Naturally, D(m, n) can be represented as a Cayley graph on the additive groupA set of vertices of D(m, n) is called an additive code if it forms a subgroup of this group. We construct a 3-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive 1-perfect codes in D(m, n ′ + n ′′ ) are sufficient. Additionally, two quasi-cyclic additive 1-perfect codes are constructed in D(155, 0 + 31) and D(2667, 0 + 127).