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bibkey: asgarli2021pseudopaley
authors: Shamil Asgarli; Chi Hoi Yip
year: 2024
title: The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes
doi: 10.48550/arXiv.2110.07176
url: https://arxiv.org/abs/2110.07176v5
claim: Definition 1.1 defines pseudo-Paley graphs; Conjecture 5.9 classifies two-dimensional cliques containing one in PP(p^4,p+1,I); Proposition 5.10 gives a conditional implication to Conjecture 5.8.
strata_touched:
  - D5/S3/Combinatorics/PseudoPaley/TwoDimensionalCliques
license: citation-only
triage: anchor

Verified locator

DOI: 10.48550/arXiv.2110.07176

Source: https://arxiv.org/abs/2110.07176v5

The v5 source is Finite Fields and Their Applications 99 (2024), 102492. The first preprint appeared in 2021. The v5 title above differs from the initial preprint title, Van Lint–MacWilliams’ conjecture and maximum cliques in Cayley graphs over finite fields.

Source statements

Cyclotomic classes (p. 1):

Let $N\mid (q-1)$. Let $C_0$ be the subgroup of ${\mathbb{F}}_q^{\ast }$ with index $N$, and let $C_1,\dots ,C_{N-1}$ be all the cosets of $C_0$, where $C_j=g^j{C}_0$. The sets $C_0,C_1,\dots ,C_{N-1}$ are called the $N$-th cyclotomic classes of ${\mathbb{F}}_q$.

Definition 1.1 (pp. 1–2):

Suppose $q$ is a prime power, $d$ a positive integer such that $2d\mid (q-1)$, and $I=\{m_1,\dots ,m_d\}\subset \{0,1,\dots ,2d-1\}$ with $|I|=d$. Let $C_0,C_1,\dots ,C_{2d-1}$ be the $2d$-th cyclotomic classes of ${\mathbb{F}}_q$. The graph $PP(q,2d,I)$ is defined to be the Cayley graph $\mathrm{Cay}({\mathbb{F}}_q^{+},D)$ where $D={\bigcup }_{j=1}^d{C}_{m_j}.$

Conjecture 5.9 (p. 17):

Let $V$ be a $2$-dimensional subspace in ${\mathbb{F}}_{p^4}$, such that $1\in V$. Then $V$ is a clique in $PP(p^4,p+1,I)$ for some $I$ if and only if $V={\mathbb{F}}_p\oplus a{\mathbb{F}}_p$, where $a=g^{(p+1)k}$ and $k$ is an odd integer.

Proposition 5.10 (pp. 17–18):

Conjecture 1.4, Conjecture 5.6, and Conjecture 5.9 together imply Conjecture 5.8.

The Proposition’s proof counts $(p^2+1)/2$ subspaces of the indicated form. That count and the implication to Conjecture 5.8 are literature statements; the Lean classification does not certify a general counting theorem or prove Conjectures 1.4, 5.6, or 5.8. Proposition 5.7 is a separate density proposition.

Encoding and scope

The formal field is GaloisField p 4, with prime field ZMod p. The cyclotomic class is the literal set of powers g ^ (j + (p+1)*m) for natural m. The graph is SimpleGraph.fromRel of the difference-in-connection relation. The permitted index sets lie in Finset.range (p+1) and have cardinality Nat.div (p+1) 2. A two-dimensional subspace containing one is a clique for some such set exactly when it is Submodule.span (ZMod p) {1, g ^ ((p+1)*k)} for an odd natural index k. Natural odd indices represent the same powers as odd integer indices because the generator order is even.

The class-map argument uses the norm power $(b+t{)}^{p^2+1}$ and its quadratic expansion. Nonconstant projective fibers have at most two points. Cliquehood forces paired fibers; the norm of the selected generator power is a square exactly when the index is even. The internal projective line is Option (ZMod p), with none representing infinity and the mate exchanging infinity with zero.

The related Xiong–Yip theorem (arXiv:2604.04126v1, Theorem 1.8, with Corollary 1.9) assumes $q\equiv -1(\mathrm{mod}d)$. Here $q=p^2$ and $d=p+1$ give $q\equiv 1(\mathrm{mod}d)$, so that result does not supply this classification. Its application and the absence of a prior settlement remain subject to the literature readings in the problem dossier.