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arxiv: 2601.12393 · v3 · submitted 2026-01-18 · 💻 cs.IT · math.CO· math.IT

2-quasi-perfect Lee codes and abelian Ramanujan graphs: a new construction and relationship

Shohei Satake This is my paper

Pith reviewed 2026-05-16 13:52 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.IT
keywords Lee codesquasi-perfect codesRamanujan graphsfinite fieldsgenerating setsp-ary codesCayley graphs
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The pith

A new infinite family of 2-quasi-perfect Lee codes is constructed from the cubic generating set in finite fields and linked to abelian Ramanujan graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives an explicit construction for an infinite family of 2-quasi-perfect p-ary Lee codes. The codes have length (q-1)/2 and dimension (q-1)/2 - 2k for q equal to a prime power at least 14 with odd prime base. They are built using the set of points (a, a cubed) inside the additive group of a quadratic extension of the finite field. The work also shows how earlier constructions of such codes correspond to known abelian Ramanujan graphs, creating a common algebraic description for both objects.

Core claim

This paper presents a new explicit infinite family of 2-quasi-perfect p-ary Lee codes of length (q-1)/2 and dimension (q-1)/2 - 2k for q = p^k >=14, p>=5 prime. The codes are derived from the generating set H_q = {(a, a^3) | a in F_q^*} of the additive group of F_{q^2}. The paper further bridges 2-quasi-perfect Lee codes constructed previously to well-known abelian Ramanujan graphs such as Li's graphs and finite Euclidean graphs.

What carries the argument

The generating set H_q of pairs (a, a^3) for a nonzero in the base field, which serves as the connection set for both the Lee code and the associated Ramanujan graph.

Load-bearing premise

The specific generating set H_q produces codes that meet the exact sphere-packing bound for 2-quasi-perfectness in the Lee metric for every such q and k.

What would settle it

Computing the Lee distance distribution or the number of covered syndromes for q=25 and k=1 and checking if it exactly matches the 2-quasi-perfect condition would falsify the claim if the counts do not align.

read the original abstract

This paper presents a new explicit infinite family of 2-quasi-perfect $p$-ary Lee codes of length $\frac{q-1}{2}$ and dimension $\frac{q-1}{2}-2k$ for $q = p^k \ge 14$, $p\geq 5$ a prime. Our codes are derived from the generating set $H_q = \{(a, a^3) \mid a \in \mathbb{F}_q^*\}$ of the additive group of the finite field $\mathbb{F}_{q^2}$. Furthermore, we bridge between 2-quasi-perfect Lee codes constructed by Mesnager, Tang, and Qi and well-known abelian Ramanujan graphs, specifically Li's graphs and finite Euclidean graphs, providing a unified theoretical framework for these families.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs a new explicit infinite family of 2-quasi-perfect p-ary Lee codes of length (q-1)/2 and dimension (q-1)/2 - 2k for q = p^k ≥ 14 with p ≥ 5 prime, using the generating set H_q = {(a, a^3) | a ∈ F_q^*} of the additive group of F_{q^2}. It further relates these codes to abelian Ramanujan graphs (Li's graphs and finite Euclidean graphs) to provide a unified framework connecting the two families.

Significance. If the construction and the 2-quasi-perfect property are rigorously established, the work supplies an explicit infinite family of Lee codes with controlled covering radius and unifies previously separate constructions in coding theory and graph theory. The parameter-free algebraic origin of H_q and the explicit link to Ramanujan graphs would constitute a concrete advance for both fields.

major comments (1)
  1. [§3, Theorem 3.2] §3, Theorem 3.2 and the subsequent counting argument: the claim that every nonzero element of F_{q^2} is represented exactly the required number of times by sums of at most two elements of H_q (or their negatives) is load-bearing for the 2-quasi-perfect property. The manuscript must supply a uniform character-sum or direct counting argument that holds independently of whether 3 divides q-1; the current treatment does not explicitly separate the cases q ≡ 1 mod 3 and q ≢ 1 mod 3, leaving the result open for half the stated parameter range (e.g., q=25 versus q=49).
minor comments (2)
  1. [§2] The notation for the Lee metric and the precise definition of quasi-perfection should be restated once in §2 for readers coming from the graph-theoretic side.
  2. [Figure 1] Figure 1 caption should indicate the precise values of q and k used for the plotted graphs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for a more explicit treatment of the counting argument in the proof of Theorem 3.2. We address the comment below and will revise the manuscript to incorporate a uniform argument.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2 and the subsequent counting argument: the claim that every nonzero element of F_{q^2} is represented exactly the required number of times by sums of at most two elements of H_q (or their negatives) is load-bearing for the 2-quasi-perfect property. The manuscript must supply a uniform character-sum or direct counting argument that holds independently of whether 3 divides q-1; the current treatment does not explicitly separate the cases q ≡ 1 mod 3 and q ≢ 1 mod 3, leaving the result open for half the stated parameter range (e.g., q=25 versus q=49).

    Authors: We agree that the current exposition of the counting argument does not explicitly separate or uniformly treat the cases according to the divisibility of q-1 by 3. In the revised manuscript we will replace the existing argument with a uniform character-sum evaluation. Specifically, the number of solutions to the relevant equations x + y = c with x, y in H_q union -H_q will be expressed via sums involving the cubic multiplicative character of F_q; these sums admit a closed-form expression (via the known values of Gauss sums for characters of order dividing 3) that is independent of whether 3 divides q-1. The resulting exact count confirms that every nonzero element of F_{q^2} is represented the required number of times for all q = p^k >= 14 with p >= 5 prime. This revision closes the gap for the full parameter range while preserving the algebraic origin of the construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit algebraic construction from finite-field generating set

full rationale

The paper defines the codes directly via the explicit generating set H_q = {(a, a^3) | a ∈ F_q^*} inside the additive group of F_{q^2} and claims the 2-quasi-perfect property follows from the distribution of sums of at most two elements of H_q. This is an algebraic counting argument over finite fields, not a self-definition, fitted parameter renamed as prediction, or reduction to a prior self-citation. The bridge to Li's graphs and finite Euclidean graphs is presented as a relationship between independently known families rather than a load-bearing justification. No equation or step reduces by construction to its own inputs; the derivation remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on standard properties of finite fields and additive groups; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Finite fields F_{q^2} exist and support the additive group structure with the map a to a^3 being well-defined for a in F_q^*
    Invoked to define the generating set H_q that produces the codes.

pith-pipeline@v0.9.0 · 5432 in / 1173 out tokens · 35994 ms · 2026-05-16T13:52:43.467361+00:00 · methodology

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Reference graph

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