Pith Number

pith:LIS6DBFK

pith:2026:LIS6DBFKRZLCFBGT6QASVETVMI

not attested not anchored not stored refs resolved

Orthogonal Polynomials and the MacWilliams Transform for Permutation-Invariant Qudit Codes

Ian Teixeira

The MacWilliams transform for permutation-invariant qudit codes equals a finite Racah transform built from orthogonal polynomials.

arxiv:2605.15372 v1 · 2026-05-14 · quant-ph · cs.IT · math.IT

Open paper page JSON Open Graph Bundle Merged state What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{LIS6DBFKRZLCFBGT6QASVETVMI}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

The intrinsic MacWilliams matrix for permutation-invariant qudit codes is identified with a finite Racah transform whose entries are given by a terminating hypergeometric series and whose rows are Racah orthogonal polynomials with parameters determined by block length and local dimension.

C2weakest assumption

The decomposition of the conjugation action on the operator space is multiplicity-free, allowing the intertwiner algebra to be identified directly with the Racah algebra without additional multiplicity factors.

C3one line summary

Derives closed-form MacWilliams matrix for permutation-invariant qudit codes as Racah polynomials with parameters set by block length and dimension.

References

28 extracted · 28 resolved · 1 Pith anchors

[1] A theorem on the distribution of weights in a systematic code, 1963

[2] F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes. North-Holland, 1977 1977

[3] Delsarte,An Algebraic Approach to the Association Schemes of Coding Theory 1973

[4] E. Bannai and T. Ito,Algebraic Combinatorics I: Association Schemes, ser. Mathematics Lecture Note Series. Menlo Park, CA: Benjam- in/Cummings, 1984 1984

[5] Quantum analog of the macwilliams identities for classical coding theory, 1997

Receipt and verification

First computed	2026-05-20T00:00:55.075628Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5a25e184aa8e562284d3f4012a92756232cdeff46d5ee10ba3e54e58518ff704

Aliases

arxiv: 2605.15372 · arxiv_version: 2605.15372v1 · doi: 10.48550/arxiv.2605.15372 · pith_short_12: LIS6DBFKRZLC · pith_short_16: LIS6DBFKRZLCFBGT · pith_short_8: LIS6DBFK

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LIS6DBFKRZLCFBGT6QASVETVMI \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 5a25e184aa8e562284d3f4012a92756232cdeff46d5ee10ba3e54e58518ff704

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "4244b728027f19aedba0ee055dfb08a22f43800b5f18e6d10860744fd0c19e10",
    "cross_cats_sorted": [
      "cs.IT",
      "math.IT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "quant-ph",
    "submitted_at": "2026-05-14T19:58:50Z",
    "title_canon_sha256": "5d7310676263a02ca69b1592dd87eb59cb09abc8636bf389a340982f644d4a04"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15372",
    "kind": "arxiv",
    "version": 1
  }
}