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We are asked to find the generating idempotents of duadic codes of length $n = 23$ over $\mathbb{F}_3$. Let’s carefully solve this step by step manually, following the usual theory for duadic codes. --- ### Step 1: Background on duadic codes * Let $n = 23$ and the field be $\mathbb{F}_3$. * A duadic code is a cyclic code of length $n$ over $\mathbb{F}_q$ whose defining set is a partition of the set of nonzero integers modulo $n$ into two disjoint sets $S_1$ and $S_2$, related by a multiplier $\mu$ (usually a quadratic residue). * Duadic codes exist over $\mathbb{F}_q$ if $q$ is a quadratic residue modulo $n$. --- ### Step 2: Check existence condition * $n = 23$ is prime. * The field is $\mathbb{F}_3$, so $q _________ ____ _______ ________ ______ ____ __________.
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