We study double-circulant codes over a class of semi-local rings arising from the idempotent construction R=Zp2+uZp2, where u2=u, and p is an odd prime. Although both algebraic settings considered admit this presentation, they correspond to two distinct rings depending on whether the additional relation pu=0 is imposed or not. These two configurations induce different ideal lattices and symmetry properties, which play a decisive role in the structure and enumeration of codes. Exploiting the Chinese Remainder Theorem, we describe self-dual and linear complementary dual (LCD) double-circulant codes in a unified, componentwise manner. Exact enumeration formulas are derived by reducing the corresponding duality constraints to norm equations over finite fields and unramified Galois extensions of Zp2. We further construct explicit Fp-linear Gray maps from R2n to Fp6n in the degenerate case pu=0 and to Fp8n in the standard case pu≠0, and show that these maps preserve self-duality and the LCD property. Assuming a standard primitive-root hypothesis on the code length, as predicted by Artin’s primitive root conjecture, we establish asymptotic existence bounds for the Gray images of both LCD and self-dual double-circulant codes via a probabilistic argument. The degenerate case pu=0 yields a shorter Gray expansion and a stronger self-dual entropy threshold, while the case pu≠0 leads to a larger self-dual ensemble with distinct asymptotic characteristics.
Saif et al. (Fri,) studied this question.