Generalised Trace-Kernel Constructions of MDS Symbol–Pair Codes over Multi-Nilpotent Chain Rings

Let Fq be a finite field, and let R denote a multi–nilpotent extension ring of theformR = Fq + uFq + vFq + wFq + u2Fq + uvFq + vwFq + w2Fq,where u3 = v3 = w3 = 0 along with commutativity and the annihilation conditionsthat ensure nilpotency of higher-order products. The additive decompositionR = Fq ⊕ N , N = ⟨u, v, w⟩,which leads to a nontrivial interaction between the field component and the nilpotent ideal. We define a generalized trace mapping of the formT (x) = x + xq + u(xq − x) + v(xq2 − x)We then study the algebraic structure of its kernelT0 = {x ∈ R | T (x) = 0}It is shown that elements of T0 generate Frobenius-stable subsets whose orbit behaviour differs from that of classical quadratic extensions.By selecting consecutive elements from T0, generator polynomialsφλ(x) =λ−3Yt=0(x − ϑt)are constructed, and it is observed that these polynomials lie in Fq[x] despite beingdefined over R. These polynomials generate cyclic and constacyclic codesXλ ⊆ RNwhich satisfy∆pair = N − K + 2,attaining the Singleton-type bound in the symbol–pair metric.