Is there a generator theory for repeated root cyclic codes over finite fields? In other words is the ring $GF(q)[x]/(x^n-1)$ principal when $(n,q)>1$?
2026-03-31 03:26:39.1774927599
The algebraic structure of repeated root cyclic codes
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I think so. The ideals of $GF(q)[x]/(x^n-1)$ are in bijective correspondence with those ideals of the polynomial ring $GF(q)[x]$ that contain the polynomial $x^n-1$. The latter are all principal ideals, because all ideals of $GF(q)[x]$ are. Furthermore, the lowest degree non-zero polynomial still generates the ideal.