Orthogonal Polynomials and the MacWilliams Transform for Permutation-Invariant Qudit Codes
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Quantum Physics arXiv:2605.15372 (quant-ph) [Submitted on 14 May 2026] Title:Orthogonal Polynomials and the MacWilliams Transform for Permutation-Invariant Qudit Codes Authors:Ian Teixeira View a PDF of the paper titled Orthogonal Polynomials and the MacWilliams Transform for Permutation-Invariant Qudit Codes, by Ian Teixeira View PDF HTML (experimental) Abstract:We derive an explicit formula for the intrinsic MacWilliams transform for permutation-invariant qudit codes. Such codes naturally live in symmetric power representations, where the relevant error sectors are determined by the irreducible decomposition of the conjugation action on the associated operator space. Using the multiplicity-free structure of this decomposition and the corresponding intertwiner algebra, we identify the intrinsic MacWilliams matrix with a finite Racah transform. The entries are given by a terminating hypergeometric series, and the rows of the matrix are Racah orthogonal polynomials with parameters determined explicitly by the block length and local dimension. Computing the spectrum of the degree-one twirl reveals that this spectrum lies on an affine quadratic lattice. Then we derive a tridiagonal multiplication rule from the representation theory of the adjoint sector. As consequences, we obtain closed-form orthogonality, detailed-balance, and involutivity identities for the transform. The resulting formula supplies an explicit MacWilliams matrix for computing linear programming bounds on permutation-invariant qudit codes. Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT) Cite as: arXiv:2605.15372 [quant-ph] (or arXiv:2605.15372v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.15372 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Ian Teixeira [view email] [v1] Thu, 14 May 2026 19:58:50 UTC (36 KB) Full-text links: Access Paper: View a PDF of the paper titled Orthogonal Polynomials and the MacWilliams Transform for Permutation-Invariant Qudit Codes, by Ian TeixeiraView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: cs cs.IT math math.IT References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)