Double circulant self-dual and LCD codes over Galois rings

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This paper investigates the existence, enumeration and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic p² and order p⁴ with p and odd prime. When p ≡ 3 4, we give an algorithm to construct a duality preserving bijective Gray map from such a Galois ring to Zₚ₂². Using random coding, we obtain families of asymptotically good self-dual and LCD codes over Zₚ₂, for the metric induced by the standard Fₚ-valued Gray maps.