Innovation
Methods, ideas, or system contributions that make the work stand out.
symbol-pair distance
repeated-root constacyclic codes
finite chain rings
maximum distance separable (MDS)
quadratic character
On symbol-pair distance of repeated-root constacyclic codes of length $4p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$
🤖 AI Summary
This study investigates the symbol-pair distance distribution and maximum distance separable (MDS) properties of repeated-root constacyclic codes of length $4p^s$ over the finite chain ring $\mathbb{F}_{p^m} + u\mathbb{F}_{p^m} + u^2\mathbb{F}_{p^m}$. By classifying the shift constant $\Delta$ according to its quadratic character—whether it is a square or a non-square—and leveraging the ideal structure, direct sum decomposition, and local ring reduction techniques, the work provides the first complete characterization of symbol-pair distances for such codes. The main contributions include deriving exact symbol-pair distance formulas for eight distinct classes of ideals, proving that only trivial ideals attain the symbol-pair Singleton bound, thereby ruling out the existence of nontrivial MDS codes, and validating these theoretical findings through an explicit example of length 20.
📝 Abstract
This paper completely determines the symbol-pair distance distributions of all repeated-root $Δ$-constacyclic codes of length $4p^{s}$ over the finite commutative chain ring $R_{3}=\mathbb{F}_{p^{m}}[u]/\langle u^{3}\rangle$, where $p^{m}\equiv1 \pmod 4$. The distance characterization is explicitly classified according to the quadratic character of the shift unit $Δ\in R_{3}^{*}$. When $Δ$ is a non-square unit, the exact symbol-pair distances are established across all eight distinct ideal classifications of the ambient ring. Conversely, when $Δ$ is a square unit, the distance profiles are derived by evaluating direct sum decompositions and local ring reductions. By evaluating the symbol-pair singleton bound, we prove that only the trivial ideal $\mathcal{C}=\langle1\rangle$ achieves maximum distance separability (MDS) , as structural constraints rule out any non-trivial MDS configurations. Finally, computational examples of length 20 over $\mathbb{F}_{5}+u\mathbb{F}_{5}+u^{2}\mathbb{F}_{5}$ are provided to validate the derived distance formulas.
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