On lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$

Daohua Wang; Ka Hin Leung; Ran Tao; Tao Zhang

REVIEW 3 major objections 5 minor 29 references

On lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For two-coordinate limited-magnitude error balls, lattice tilings of $\mathbb{Z}^n$ exist only up to an explicit dimension bound when $k_1+k_2+1$ is composite; $\mathcal{B}(n,2,3,0)$ tiles exactly at $n=3$, and $\mathcal{B}(n,2,k,k-1)$…

desk verdict The two classification theorems look solid and interesting, but the general non-existence proof collapses on a counting error in equation (5), so Theorem 1.3 as written is unsupported. read the letter →

arxiv 2505.08495 v1 pith:JJNHMU2A submitted 2025-05-13 math.CO cs.ITmath.IT

Ka Hin Leung , Ran Tao , Daohua Wang , Tao Zhang This is my paper · ORCID

classification math.COcs.ITmath.IT MSC 52C2211H3111H71

keywords latticetilingperfectcodelimitedmagnitudeerrorballgroupringfiniteabeliantwo-coordinateerrorsflashmemorynon-existencethreshold

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks when the integer lattice $\mathbb{Z}^n$ can be tiled by the limited-magnitude error ball $\mathcal{B}(n,2,k_1,k_2)$ of vectors with at most two nonzero coordinates, each coordinate lying in $[-k_2,k_1]$. Because a lattice tiling is exactly a linear perfect code for this asymmetric error model, the question carries coding-theoretic weight, especially for flash memory where symbol values are integer vectors. The paper's main theorem states that whenever $k_1+k_2+1$ is composite, no such lattice tiling exists once $n$ exceeds an explicit threshold quadratic in $k_1-k_2$. It also completely settles two families: $\mathcal{B}(n,2,3,0)$ tiles $\mathbb{Z}^n$ if and only if $n=3$, and $\mathcal{B}(n,2,k,k-1)$ tiles no $\mathbb{Z}^n$ for any $n\ge 3$. The proof translates tilings into decompositions of finite abelian groups and then bounds the overlaps of the resulting power-product sets.

What carries the argument

The machine is the group-ring decomposition $G=e+\sum_{i\in[-k_2,k_1]^*} T^{(i)} + \sum_{i\le j} S(i,j)$ together with the counting functions $\psi(m,i,j)=|\{t\in T:t^m\in S(i,j)\}|$ and their one-coordinate analogues. The pivotal component is Lemma 2.9, which bounds the maximum multiplicity $C$ of the power map $t\mapsto t^{k_1+k_2+1}$ on the generator set $T$ by $3\sqrt{p}$ (or $3$ when $p=2$, and $4$ when $k_1+k_2=3$). The proof embeds the relevant $T$-powers into a maximal elementary abelian $p$-subgroup $H$ and counts inside one coset of $H$, using the order bound $|H|\le p^2$. This $C$ bound converts into uniform $O(n)$ bounds on all set intersections, turning the tiling condition into a quadratic inequality in $n$ that cannot hold once $n\ge \lfloor B/A\rfloor+1$.

What would settle it

Re-run the computational search claimed for $\mathcal{B}(n,2,3,0)$ at $n=4,5,6$; any tiling found would refute Theorem 1.1, and because the proof's inequality (3) together with Lemma 2.9 rules out $n\ge 7$, the search is the decisive check. For the general theorem, compute the collision count $C=\max_g|\{h:g^{k_1+k_2+1}=h^{k_1+k_2+1}\}|$ in a group whose elementary abelian $p$-core has order exceeding $p^2$; a value above $3\sqrt{p}$ would invalidate Lemma 2.9(a) and with it inequality (9).

Watch

Extended reading notes

Core claim

The central discovery is a structural constraint on any finite abelian group $G$ and $n$-subset $T$ that realizes a tiling: every element of $G$ must be uniquely the identity, a single power $t^i$ with $i\in[-k_2,k_1]^*$, or a product of two such powers from distinct elements of $T$. The paper expresses $|G|$ as a sum of the sizes of these sets $T^{(i)}$ and $S(i,j)=\{g^ih^j:g,h\in T,\,g\neq h\}$, and bounds the intersections among these sets using the scarcity of collisions of the map $g\mapsto g^{k_1+k_2+1}$. When $k_1+k_2+1$ is composite with smallest prime divisor $p$, a key lemma caps the maximum collision multiplicity by $3\sqrt{p}$, and the resulting inclusion-exclusion lower bound on the union of $S$-sets exceeds $|G|$ for large $n$, forcing non-existence above an explicit threshold. The two complete classifications follow by refining the index sets and checking the finite range below the threshold.

Load-bearing premise

Everything hinges on the claim that, whenever $k_1+k_2+1$ is composite with smallest prime divisor $p$, at most $3\sqrt{p}$ elements of the generator set $T$ can have the same $(k_1+k_2+1)$-th power in the tiling group; the proof of this claim is the paper's most compressed step, and if it fails, inequality (9) collapses.

Editorial extensions

If this is right

For $k_1=3, k_2=0$, a lattice tiling of $\mathbb{Z}^n$ exists if and only if $n=3$, realized explicitly by $T=\{1,10,26\}$ inside the cyclic group $\mathbb{Z}_{37}$.
For every $k\ge 2$ and every $n\ge 3$, $\mathbb{Z}^n$ admits no lattice tiling by $\mathcal{B}(n,2,k,k-1)$.
Whenever $k_1+k_2+1$ is composite, no lattice tiling of $\mathbb{Z}^n$ by $\mathcal{B}(n,2,k_1,k_2)$ exists for $n\ge 2(k_1-k_2)^2+12(k_1-k_2)+2k_2+8$.
The dimension threshold is quadratic in $k_1-k_2$, so for fixed $k_2$ the possible tiling dimensions grow quadratically with the asymmetry gap.
When $k_1+k_2+1$ is prime, the method does not apply if the group order is a $p$-power; the paper notes that for $k_1+k_2+1=5$ only $n=2$ was found by computation, leaving the prime case as the open frontier.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The composite condition is likely an artifact of the proof technique: a natural conjecture is that for every $k_1>k_2\ge 0$ with $k_1+k_2\ge 3$ there are only finitely many $n$ admitting a lattice tiling, with prime values of $k_1+k_2+1$ needing a different collision argument.
The explicit three-dimensional tiling of $\mathcal{B}(n,2,3,0)$ hints that small sporadic tilings exist precisely when the group order is small; testing dimensions near $2(k_1-k_2)^2$ for small $k_1,k_2$ could reveal further isolated examples.
The bound $C\le 3\sqrt{p}$ is loose; a sharper collision bound would improve inequality (9) and could shrink the thresholds substantially, possibly covering cases currently outside the theorem's reach.
The same group-ring skeleton with two-coordinate index sets should carry over to $t\ge 3$ by replacing pairs with $t$-tuples, with the collision map remaining $g\mapsto g^{k_1+k_2+1}$; the authors state this extension as planned future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this paper has two genuinely new classification results that look like real progress, and one general theorem whose proof does not hold up. The classification of B(n,2,3,0) (tiling iff n=3) and the full resolution of B(n,2,k,k-1) are both substantive and appear plausible. The index-set counting technique used in those sections is clever and worth reading. I did not find a concrete flaw in Sections 3 and 4.

The problem is Section 5. The stress-test note flags a counting error, and I think the core of it is correct. Equation (5) treats the index set Z as having (k1-k2)^2/2 elements, but with the displayed definition Z={(i,j): k2+1 ≤ i ≤ j ≤ k1} the count is d(d+1)/2 where d=k1-k2. That is already a red flag. Then |X1|, the part of X described by i∈[-k1,-k2-1], j∈[k2+1,k1] with j-i≤k1+k2+1, is also d(d+1)/2—the same as |Z|. The extra set difference comes only from X2, which has 2Mk2 elements. So replacing Z by X changes the number of S-sets by 2Mk2, not by A/2 = [k1+(4M-1)k2]/2. For k2=0 the change is exactly zero. Inequality (9) claims a positive n^2 term on the left from this replacement, but the only n^2 term available from the definitions is 2Mk2(n^2-n). For k2=0 there is no n^2 term at all. The stress-test's secondary claim about omitted cross pairs is wrong—Y as defined does include the pairs with i∈[-k2,k2] and j∈[k2+1,k1]—but that does not rescue (9).

The O(n) intersection bounds in Lemmas 5.4–5.11 may be fine, but they cannot compensate for a missing n^2 term. As written, the proof of Theorem 1.3 and Corollary 1.4 is unsupported. Separately, the computational checks for n=4,5,6 and n=3,k=2 are undocumented; that is a verification gap, but minor compared with the Section 5 issue.

Who this is for: people working on lattice tilings, perfect codes, and limited-magnitude error models. The first two theorems are worth publishing on their own if the Section 5 problem is fixed or the general theorem is withdrawn. I would send this to a serious referee, but ask the referee to focus on the counting in Section 5. If the authors can correct the miscount and still get a non-existence theorem, fine; if not, the paper should be split and Theorem 1.3 removed.

Referee Report

3 major / 5 minor

Summary. The paper studies lattice tilings of Z^n by asymmetric limited-magnitude error balls B(n,2,k1,k2) for k1>k2. Using the Horak--AlBdaiwi characterization of lattice tilings by finite abelian groups, it develops a group-ring and counting framework and claims three main results: Theorem 1.1 completely classifies B(n,2,3,0) tilings (they exist only for n=3); Theorem 1.2 excludes all B(n,2,k,k-1) tilings for n>=3, k>=2; and Theorem 1.3, together with Corollary 1.4, shows that for composite k1+k2+1 no lattice tiling exists in all sufficiently high dimensions. The proofs proceed by assuming a tiling, passing to a finite abelian group G of the correct order, and deriving lower bounds on the size of a modified union of S(i,j) sets. The final dimension bound is obtained by comparing the set X with the set Z in Section 5.

Significance. If the results are correct, they settle two natural infinite families and provide a general non-existence mechanism for a class of asymmetric error balls, which is a substantive advance over the existing partial results. The reduction to finite group counting is elegant, and the explicit generator T={1,10,26} in Z_37 for the n=3, (3,0) tiling is a valuable concrete certificate. The paper also gives a clear roadmap for the general composite case. However, verification is currently hindered by an incomplete proof of the key collision bound (Lemma 2.9), by undocumented computer checks for several small cases, and by a very compressed algebraic derivation of the main inequality (9). These issues are local in nature but load-bearing for the stated theorems.

major comments (3)

[Section 2, Lemma 2.9] The bound C<=3*sqrt(p) in Lemma 2.9(a) is load-bearing: it enters Lemma 2.10(c) and, through inequality (9), the proof of Theorem 1.3. The proof as written is not complete. The step 'As |m-(|x|+y)ell|<2(ell-1), we conclude that |x|+y=p-1' requires an additional argument that k1+k2+1-(|x|+y)ell is a positive multiple of ell and hence equals ell; the displayed inequality alone does not imply the conclusion. More importantly, the inequality ((p-1 choose 2)|A_i|(|A_i|-1)<=p^2 appears to treat the displayed union as a subset of a coset of H with pairwise disjoint full-size contributions, but no proof of disjointness or of the stated size of the expressions A_i^{(alpha)}A_i^{(beta)}-A_i^{(alpha+beta)} is given. The same kind of unproved counting is used in part (b) for the bound on psi(m,0). Please provide a complete proof or a precise reference for this lemma.
[Section 3 and Section 4] The complete classifications in Theorems 1.1 and 1.2 depend on undocumented computational searches. Section 3 states 'For 3<=n<=6, we use a computational search and check that no tiling of Z^n exists by B(n,2,3,0) if n=4,5 and 6', and Section 4 states 'If k=2, a computational search confirms that no lattice tiling of Z^3 by B(3,2,2,1) exists.' No algorithm, source code, or verifiable certificate is provided, so the reader cannot check these finitely many cases. Because these checks are essential to the claimed full classifications, the authors should supply reproducible code or a complete hand-checkable mathematical verification for these instances.
[Section 5, inequality (9)] The derivation of the key inequality (9) from Lemmas 5.9, 5.10, 5.11 and inequalities (7)-(8) is a large unshown algebraic step. In particular, the assembly of the constant B and the transition to the final bound n>=floor(B/A)+1 are not displayed. Since Theorem 1.3 rests entirely on this inequality, the authors should present the intermediate inequalities and show explicitly how each term of B arises. Relatedly, equation (5) is correct only if one uses |S(i,i)|=n(n-1)/2 for the d diagonal terms; the text appears to count d(d+1)/2 pairs each of size n^2-n, so this step must be written out.

minor comments (5)

[Section 5, equations (5) and (9)] I checked the specific count objection raised about the leading term of (9) and do not find it to be valid. Although the number of pairs in Z is d(d+1)/2 with d=k1-k2 rather than d^2/2, the displayed total (k1-k2)^2/2 (n^2-n) is correct because the d diagonal pairs S(i,i) have size n(n-1)/2, not n^2-n. Likewise, when k2=0 the replacement of Z by X does not add new pairs, but it replaces d diagonal half-size sets by d off-diagonal boundary sets of size roughly n^2-n, which produces the positive leading term k1/2(n^2-n). The text should spell this out, since the current wording invites the miscount.
[Section 4, Lemma 4.1] In Lemma 4.1(a), the equality '|S(-k,1)|=|S(-k,1)|=n^2-n' contains a self-referential typo; it should presumably compare |S(-k,1)| with |S(1,-k)| or another symmetric partner.
[Section 5, Lemma 5.4] In the proof of Lemma 5.4, the summation 'Sum_{j in [k2+1,k2]*}' has an empty range and should presumably be '[k2+1,k1]'.
[Section 5, Lemma 5.9] In the proof of Lemma 5.9, the symbol 'phi(ell,-i_1,j_2)' should be 'psi(ell,-i_1,j_2)'.
[Corollary 1.4] The displayed bound contains a stray period inside the formula: '2k2 + 8.' should read '2k2 + 8'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning: the tiling nonexistence proofs reduce to an external group-ring characterization and independent counting arguments; self-citations are background only.

full rationale

I walked the derivation chain from the main theorems back to their inputs. Theorem 1.3 and Corollary 1.4 rest on inequality (9), obtained by comparing the cardinality of the set G' (built with index set X) with |G| (built with Z). The comparison uses Lemmas 5.1-5.11 and Lemma 2.9, all of which are counting statements about the hypothetical group G and subset T; none assumes the nonexistence conclusion being proved. Theorem 2.3, the only external structural input, is the Horak-AlBdaiwi criterion [6], an independent theorem stating that a lattice tiling exists iff a finite abelian group G of the stated order and an n-subset T satisfy a bijection condition. This is not a restatement of the paper's target result. Lemma 2.9 is proved in-paper by a containment count in a maximal elementary p-subgroup; even though the proof is compressed, it is not a disguised version of any main theorem. The self-citations, including [28] on B(n,2,1,1), appear only in the introduction as related work and play no load-bearing role in Sections 3-5. No parameter is fitted to a subset of data and then renamed a prediction, and no uniqueness or existence theorem is imported from the authors' prior work to force a choice. The only concerns visible in the manuscript, such as the compressed proof of Lemma 2.9 and a possible arithmetic slip in equation (5), are correctness or robustness issues, not circularity: the claimed nonexistence is not an input to the counting. Thus I find no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on an external tiling-group equivalence and standard finite abelian group theory. There are no fitted parameters: the constants A and B in Theorem 1.3 are derived bounds, not tuned to data. The main unverified inputs are the undocumented computational searches, tracked in red flags.

assumptions (3)

standard math Theorem 2.1 of Horak-AlBdaiwi: lattice tiling exists iff there is a finite abelian group G of size |V| and a homomorphism from Z^n onto G that is bijective on V.
External theorem, cited as [6], used in Section 2 to translate the tiling problem into a group-ring equation.
standard math Standard facts on finite abelian groups: Sylow subgroups, elementary p-subgroups, and group-ring arithmetic over Z.
Invoked throughout Sections 2-5, especially in Lemmas 2.8 and 2.9.
domain assumption For a lattice tiling, the translates of the error ball partition Z^n, so the sets {e}, T^(i), S(i,j) are disjoint as stated in Theorem 2.3(2).
This is the definitional content of a tiling; it is not an extra hypothesis.

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Pith. "Pith review of On lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$." pith.science (2026). https://pith.science/paper/JJNHMU2A

@misc{pith2026250508495,
  author       = {Pith},
  title        = {Pith review of: On lattice tilings of $\mathbbZ^n$ by limited magnitude error balls $\mathcalB(n,2,k_1,k_2)$ with $k_1>k_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJNHMU2A}},
  note         = {Machine review of arXiv:2505.08495}
}

abstract

Lattice tilings of $\mathbb{Z}^n$ by limited-magnitude error balls correspond to linear perfect codes under such error models and play a crucial role in flash memory applications. In this work, we establish three main results. First, we fully determine the existence of lattice tilings by $\mathcal{B}(n,2,3,0)$ in all dimensions $n$. Second, we completely resolve the case $k_1=k_2+1$. Finally, we prove that for any integers $k_1>k_2\ge0$ where $k_1+k_2+1$ is composite, no lattice tiling of $\mathbb{Z}^n$ by the error ball $\mathcal{B}(n,2,k_1,k_2)$ exists for sufficiently large $n$.

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Buzaglo and T

S. Buzaglo and T. Etzion. Tilings withn-dimensional chairs and their applications to asymmetric codes.IEEE Trans. Inform. Theory, 59(3):1573–1582, 2013

2013

[2] [2]

Cassuto, M

Y. Cassuto, M. Schwartz, V. Bohossian, and J. Bruck. Codes for asymmetric limited- magnitude errors with application to multilevel flash memories.IEEE Trans. Inform. Theory, 56(4):1582–1595, 2010. 25

work page 2010

[3] [3]

Etzion.Perfect codes and related structure

T. Etzion.Perfect codes and related structure. World Scientific, 2022

2022

[4] [4]

Z. Guan, H. Wei, and Z. Xiang. On lattice tilings of asymmetric limited-magnitude balls B(n,2,m,m-1). arXiv: 2501.08636

arXiv

[5] [5]

Hickerson and S

D. Hickerson and S. Stein. Abelian groups and packing by semicrosses.Pacific J. Math., 122(1):95–109, 1986

1986

[6] [6]

Horak and B

P. Horak and B. F. AlBdaiwi. Diameter perfect Lee codes.IEEE Trans. Inform. Theory, 58(8):5490–5499, 2012

2012

[7] [7]

Kløve, B

T. Kløve, B. Bose, and N. Elarief. Systematic, single limited magnitude error correcting codes for flash memories.IEEE Trans. Inform. Theory, 57(7):4477–4487, 2011

work page 2011

[8] [8]

Kløve, J

T. Kløve, J. Luo, I. Naydenova, and S. Yari. Some codes correcting asymmetric errors of limited magnitude.IEEE Trans. Inform. Theory, 57(11):7459–7472, 2011

2011