Length Bounds for Irreducible Cyclic MDS Codes

Abstract

We prove the MDS length bound for nontrivial irreducible cyclic codes of every composite length. The classical trace realization turns the MDS property into independence of elements of a multiplicative subgroup. Every nonidentity subgroup must have the full extension degree. This restriction yields length congruences, and the Frobenius-orbit description of the coefficient field gives an exact descent obstruction. A cyclotomic argument reduces each fixed dimension to finitely many parameters. For dimensions at most thirty, established geometric bounds reduce the alphabet to a small finite range; the remaining twelve parameter triples are excluded by explicit products of at most three determinant factors and a low-degree multiplier argument. An independent argument proves the composite-length theorem in arbitrary dimension. The standardness conclusion is stated separately as a conditional consequence of the classification statement of Feng, Hollmann, Li, and Xiang; the correct extension-field reduction retains a semilinear quotient. The general prime-length problem is not settled here. Neither length proof uses the automorphism classification or the recent odd-characteristic range announced by Fan.