Repeated-Root Constacyclic Codes of Length nps from Half-Cyclotomic Defining Sets

Repeated-root constacyclic codes form an important class of algebraic error-correcting codes, combining the cyclic symmetry of constacyclic codes with the additional multiplicity structure produced by repeated roots. These multiplicities are usually a source of technical difficulty in distance analysis; in this paper, we use them instead as controllable design parameters. Let q=pm, let n be prime to p, and let λ∈Fq*. We study repeated-root λ-constacyclic codes of length nps obtained by lifting the half-cyclotomic defining sets. Writing λ=λ0ps gives xnps−λ=(xn−λ0)ps; hence every code is determined by a multiplicity profile e:Γ→0,1,…,ps on the set Γ of q-cyclotomic coset leaders in Zn,r, where r=ord(λ0). Equivalently, e determines a nested sequence of simple-root defining sets Tps−1⊆…⊆T0⊆Zn,r. The main result is a layerwise distance formula: if Ct is the simple-root code defined by Tt and Pt=∏j(tj+1) for the p-adic expansion t=∑jtjpj, then dH(C(e))=min0≤t<psPtdH(Ct). This reduces the repeated-root distance problem to certified distance information for the simple-root layers. Combining the formula with the half-cyclotomic bounds in the two-coset-size case gives explicit certified lower bounds for complete families of repeated-root codes. We also formulate a verified refined layer bound, prove a conditional half-rate family theorem with distance growth at least a constant multiple of N/logqN for N=nps, and give a lifting criterion for transferring certified simple-root parameters to the repeated-root setting. The construction is carried out for the length families n=(qρ−1)/(rσ), n=Φρb(q), and n=Φρ1ρ2(q).