On symbol-pair distance of repeated-root constacyclic codes of length $4p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$
2026-06-29 • Information Theory
Information Theory
AI summaryⓘ
The authors studied a specific type of error-correcting codes called repeated-root Δ-constacyclic codes over a special ring with elements involving u³=0. They fully figured out how far apart codewords are when measured by symbol-pair distance, depending on whether the defining shift unit Δ is a square or non-square in the ring. Their work shows that only the simplest code (the trivial one) can achieve the best possible error correction (MDS) under these conditions. They also gave examples with codes of length 20 to confirm their theoretical findings.
symbol-pair distanceΔ-constacyclic codesrepeated-root codesfinite commutative chain ringquadratic charactermaximum distance separability (MDS)ideal classificationslocal ringsingleton bounddirect sum decomposition
Authors
Payel Chandra, Kalyan Hansda
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.