{"id":"af788d30-d63d-4d3c-b522-f29f00b4a4e2","arxiv_id":"2412.06223","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximum size of an (n,t,w,e)-LPECC code is determined asymptotically, with new tight bounds when the error-correction strength is e=w-2 and a new exact formula for (n,t,3,1) under divisibility conditions on n.","lead":"This paper proves new upper bounds and asymptotic formulas for the maximum size of low-power error-correcting cooling codes, which protect on-chip buses from overheating while saving power and correcting errors. The results settle the exact growth rate of these codes for a wide range of bus sizes and give the exact optimum in several previously unknown cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper's main results are well-structured: the upper bound in Section 5.1 is self-contained, and the lower bound via constant-weight codes is a standard application of a cited theorem. The reader's weakest assumption—Lemma 5.6—does not appear to be a genuine vulnerability because the cited theorem, if stated correctly in [16], applies to exactly the fixed parameters used in Construction 5.4. The more plausible source of error is the compressed greedy arguments in Claims 3.4, 3.5, and 4.6, which are not fully detailed; however, I could not construct a counterexample and the algebraic steps check out in the minimal cases. Since no concrete flaw was found, the ACCEPT verdict remains appropriate, though a fuller proof of the greedy claims would reduce the residual correctness risk.","tokens_in":23374,"tokens_out":50040,"duration_ms":454725,"concrete_test":"Verify the exact statement in Liu–Shangguan [16] against Lemma 5.6: confirm that the published theorem gives A_q(n,d,w) ≥ (1−o(1)) times the Johnson bound for every fixed q, fixed odd d, and fixed w without hidden conditions such as d ≤ w. If hidden conditions exist, check whether they hold for the parameters w+t and 2(e+t)+1 under e ≤ w−1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central asymptotic claim in Corollary 5.8 rests on the upper bound of Theorem 5.1 and the lower bound obtained from Construction 5.4 together with Lemma 5.6, which quotes Liu–Shangguan [16]. Lemma 5.6 applies verbatim to Construction 5.4: with fixed q, weight w+t, and odd distance 2(e+t)+1, setting e' = e+t gives exactly the Johnson-type bound (q-1)^{w-e} C(n,w-e)/C(w+t,w-e). Because q, t, w, e are fixed as n→∞, no uniformity issue arises. The only internal soft spot is the terse greedy counting in Claims 3.4, 3.5, and 4.6; I spot-checked the small-parameter cases and found no counterexample, but a fully formal exposition would strengthen the paper. No load-bearing flaw was identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the maximum size C(n,t,w,e) of low-power error-correcting cooling (LPECC) codes, which combine thermal constraints (no transitions on the t hottest wires), power constraints (at most w transitions per transmission), and error correction (up to e errors). The main results are: (i) for w ≥ t^2+2t+2, an exact upper bound C(n,t,w,w-2) ≤ floor(binomial(n+1,2)/binomial(w+t,2)) with divisibility-based tightness and an extremal characterization via (n+1,w+t,1)-BIBDs (Theorem 3.1, Corollary 3.6, Theorem 3.7); (ii) for small weight w=3, the upper bound C(n,t,3,1) ≤ floor(n(n+1)/(6(t+1))) and matching constructions from 3-frames, yielding the asymptotic C(n,t,3,1) ~ n^2/(6(t+1)) (Theorem 4.1, Corollary 4.10); (iii) a general q-ary asymptotic result C_q(n,t,w,e) ~ (q-1)^{w-e} binomial(n,w-e)/binomial(w+t,w-e) under the condition 2 binomial(w,w-e) ≥ binomial(w-t-1,w-e)+binomial(w+t,w-e) (Corollary 5.8), proved via a Johnson-type upper bound and a lower bound from constant-weight codes; and (iv) related bounds and exact values for constant-power error-correcting cooling codes (Theorem 5.10, Corollary 5.11).","tokens_in":23478,"tokens_out":19272,"duration_ms":178102,"significance":"If the results stand, the paper substantially extends earlier work, which was limited to small t and w, to arbitrary fixed t with large w and to a general q-ary asymptotic regime. The upper-bound machinery—pair counting in Lemma 3.3, the Turán-based argument in Theorem 4.1, and the Johnson-type bound in Theorem 5.1—is coherent, and the use of external tools (Rödl's nibble, Wilson's and Keevash's existence theorems, and Liu–Shangguan's near-optimal constant-weight codes) is appropriate for the fixed-parameter setting. The extremal structure theorem (Theorem 3.7) is a strong and elegant result. The paper is essentially free of fitted parameters, and the asymptotic claims are concrete and falsifiable. The main weakness is that several load-bearing counting steps are only sketched as 'greedy' arguments, and one asymptotic derivation omits a necessary monotonicity step.","major_comments":[{"comment":"The lower bounds on τ(P_i) in Claims 3.4(c), 3.5(b), and 5.3 are proved by a 'greedy' counting sentence that is not a complete formal argument. These claims are load-bearing for the main upper bounds in Theorem 3.1 and Theorem 5.1. Please provide an explicit ordering of the vertices in Y, and prove that after removing previously used vertices, each y_j contributes at least (w-1)-j (respectively, binomial(w-j,w-e-1) in the q-ary claim) distinct previously uncounted pairs or (w-e)-subsets. In particular, the treatment of possible overlaps between the contributions for different y_j must be stated precisely.","section":"§3, Claims 3.4(c), 3.5(b); §5, Claim 5.3"},{"comment":"The statement that Corollary 4.10 follows 'immediately' from Theorem 4.1 and Theorem 4.9 is not fully justified, because Theorem 4.9 provides a lower bound only when n ≡ 2 (mod 6). To conclude the limit over all n, the authors should explicitly state and prove the zero-padding monotonicity C(n+1,t,3,1) ≥ C(n,t,3,1), or otherwise extend the construction to all sufficiently large n. This is a short argument but it is currently absent.","section":"§4, Corollary 4.10"}],"minor_comments":[{"comment":"The notation 'w ≥ t2 + 2t + 2 ≥ 5' is slightly confusing; it should be written as 'w ≥ t^2 + 2t + 2', noting that this implies w ≥ 5 for t ≥ 1.","section":"§3, statement of Theorem 3.1"},{"comment":"In the proof of Claim 4.6, the transition from the independence number bound to the Turán application is terse; please spell out that e(\\bar{H}) ≤ e(T(v2(P_i), r)) and that the complement of T(v,r) has exactly λ binom(ceil(v/r),2) + (r-λ) binom(floor(v/r),2) edges, matching Lemma 4.4.","section":"§4, Claim 4.6"},{"comment":"There is a typo: 'Note that here we all consider' should read 'Note that here we always consider'.","section":"§5.2, paragraph after Remark 5.9"},{"comment":"For completeness, please state explicitly that the o(1) in Lemma 5.6 is with respect to n for fixed q, t, w, e, since the application in Corollary 5.7 requires only this fixed-parameter uniformity.","section":"§5, Corollary 5.7"}],"recommendation":"major_revision","confidential_remarks":"I believe the mathematical results are very likely correct and the paper is a solid contribution. The recommendation of major_revision is driven by two load-bearing proof gaps that are local and fixable: the 'greedy' counting in the upper-bound claims and the missing monotonicity argument in Corollary 4.10. Once those are filled, I would expect the paper to be suitable for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Yehao, this is a genuine advance in the LPECC-code line. Prior tight bounds were stuck at w=3,4 and t<=2; this paper extends exact results to all t for (n,t,3,1), proves a general upper bound and extremal characterization for w >= t^2+2t+2 with e=w-2, and gives a q-ary asymptotic formula for a wide parameter range. The main theorems are new and correctly stated. The proof strategy is sound: pair counting, Turán, Johnson bounds, and frames all fit together, and the divisibility conditions are handled carefully. The extremal Theorem 3.7 — equality forces an (n+1,w+t,1)-BIBD — is a nice bonus. The asymptotic Corollary 5.8 is the centerpiece, and the stress-test is right that Lemma 5.6 applies verbatim to Construction 5.4, so there is no hidden uniformity gap.\n\nThe soft spots are minor. The greedy lower bounds in Claims 3.4(c), 3.5(b), and 4.6 are terse; they are the sort of step that is probably true but takes a page to write out rigorously. I spot-checked small cases and found no counterexample. The paper leans on external existence results — Rödl nibble, Keevash designs, Liu-Shangguan near-optimal constant-weight codes — but those are established and cited properly; the authors do not claim to prove them. There is no circularity and no fitted parameter. What the paper does not do is address encoding or decoding complexity, so the practical on-chip-bus story is thin. That is not a flaw for a pure coding-theory paper, but it caps the audience.\n\nThis deserves a serious referee. I would send it to review and expect acceptance after a moderate revision that expands the terse counting arguments and maybe states clearly which divisibility conditions are needed. I would cite it in my own work on packing-based bounds. You can take it to the reading group as a clean example of the double-counting + external existence template.\n\nFor peer review: accept with minor revisions.","headline":"Solid extremal-combinatorics paper that settles the asymptotic and many exact values for LPECC codes; the proofs check out and the external dependencies are legitimate.","tokens_in":24060,"tokens_out":2022,"would_cite":true,"duration_ms":18956,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B25","05B05","94B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives exact and asymptotic formulas for the maximum size of low-power error-correcting cooling codes, over a parameter range wider than any previously known.","keywords":["LPECC codes","cooling codes","low-power cooling codes","packings","Steiner systems","frames","constant-weight codes"],"falsifier":"Fix $q=2$ and parameters $(t,w,e)$ satisfying $2\\binom{w}{w-e} \\ge \\binom{w-t-1}{w-e}+\\binom{w+t}{w-e}$, and compute $A_2(n,2(e+t)+1,w+t)$ along a sequence of $n$. If for some fixed $\\varepsilon>0$ infinitely many $n$ fall below $(1-\\varepsilon)\\binom{n}{w-e}/\\binom{w}{w-e}$, the construction's lower bound fails and $C_2(n,t,w,e)$ cannot reach the claimed asymptotic value.","tokens_in":23140,"feed_emoji":"🔥","tokens_out":12284,"duration_ms":111536,"temperature":0.7,"pith_summary":"This paper establishes the largest possible size of a low-power error-correcting cooling (LPECC) code: a coding scheme for an $n$-wire bus that never switches the $t$ hottest wires, makes at most $w$ transitions per transmission, and corrects up to $e$ bit errors. The main results are exact and asymptotic formulas for that maximum, denoted $C(n,t,w,e)$. For large error budgets ($e = w-2$, with $w \\ge t^2+2t+2$) the paper proves $C(n,t,w,w-2) \\le \\lfloor \\binom{n+1}{2}/\\binom{w+t}{2}\\rfloor$, tight for large $n$ satisfying divisibility conditions; for $w=3,e=1$ it proves $C(n,t,3,1) \\le \\lfloor n(n+1)/(6(t+1))\\rfloor$, also tight infinitely often. For general $q$-ary codes satisfying a binomial-coefficient inequality, it shows $C_q(n,t,w,e) \\sim (q-1)^{w-e} \\binom{n}{w-e}/\\binom{w+t}{w-e}$ as $n$ grows. A sympathetic reader would care because these are the first tight bounds for most of the parameter range, giving the exact information-theoretic cost of simultaneously cooling, saving power, and correcting errors on a bus.","feed_headline":"Exact size caps found for power-saving error-correcting bus codes","feed_subtitle":"For large error budgets the maximum code size equals a simple binomial ratio, tight under divisibility.","key_machinery":"The machinery is the set-theoretic representation of an LPECC code as a partition $P_1 \\cup \\cdots \\cup P_b$ of blocks in $2^{[n]}$, where each block is the support of a codeword and the error-correction property forces blocks from different parts to have symmetric difference at least $2e+1$. The upper bounds come from double counting: each part must cover at least $\\binom{w+t}{2}$ pairs or, in the general case, $\\binom{w+t}{w-e}$ subsets of size $w-e$, while all parts together can cover at most $\\binom{n+1}{2}$ pairs or $(q-1)^{w-e}\\binom{n}{w-e}$ objects. The lower bounds come from a construction that turns a $q$-ary constant-weight code of weight $w+t$ and odd distance $2(e+t)+1$ into an LPECC code by cutting each support down to size $w$; the relevant distance calculation shows the error-correcting property survives after deleting $t$ coordinates from every support.","core_discovery":"On its own terms, the paper claims that the LPECC constraint set has a precise combinatorial price: the maximum number of messages is, up to a binomial ratio, the same as the maximum size of a packing or constant-weight code. Concretely, for binary codes with $e=w-2$ and $w \\ge t^2+2t+2$, every $(n,t,w,w-2)$-LPECC code has size at most $\\lfloor \\binom{n+1}{2}/\\binom{w+t}{2}\\rfloor$, and when $n$ is sufficiently large with $\\binom{w+t}{2}$ dividing $\\binom{n+1}{2}$ and $w+t-1$ dividing $n$, there exist codes meeting it; any such optimal code must be the image of an $(n+1,w+t,1)$-balanced incomplete block design. For $w=3,e=1$, the paper proves $C(n,t,3,1) = \\lfloor n(n+1)/(6(t+1))\\rfloor$ for even $n \\equiv 2 \\pmod{3(t+1)}$. For the general $q$-ary setting, whenever $2\\binom{w}{w-e} \\ge \\binom{w-t-1}{w-e}+\\binom{w+t}{w-e}$, it establishes $C_q(n,t,w,e) \\sim (q-1)^{w-e} \\binom{n}{w-e}/\\binom{w+t}{w-e}$ as $n \\to \\infty$.","pith_inferences":["A natural reading the paper does not spell out is that the LPECC constraints become asymptotically free exactly on one side of the binomial inequality, while the small-$w$ arguments point to a separate regime whose rate is $1/((t+1)w(w-1))n^2$; locating the precise boundary between these regimes is a testable extension.","The extremal characterization says optimal large-$w$ codes are exactly BIBD-induced, so known explicit constructions of balanced incomplete block designs should yield explicit optimal LPECC encoders; the paper does not develop that algorithmic direction.","Because the $q$-ary bound only uses Johnson-type counting, the same asymptotic should extend to other transition-cost models, such as energy proportional to the number of changed nonzero symbols; this is a reformulation to test rather than a claim of the paper.","The asymptotic lower bound relies on an imported existence result for near-optimal constant-weight codes, so a targeted numerical check on the exact parameters used in the construction would indicate how robust the limit is when the binomial inequality is barely satisfied."],"forward_implications":["For large $w$ relative to $t$, the exact size of the best $(n,t,w,w-2)$-LPECC code is $\\lfloor \\binom{n+1}{2}/\\binom{w+t}{2}\\rfloor$ whenever the divisibility conditions hold, and the optimal codes are exactly those induced by balanced incomplete block designs with parameters $(n+1,w+t,1)$.","For $w=3,e=1$, the maximum is $\\lfloor n(n+1)/(6(t+1))\\rfloor$ for every even $n \\equiv 2 \\pmod{3(t+1)}$, so the asymptotic density $n^2/(6(t+1))$ is attained on an infinite arithmetic progression.","For any fixed $q,t,w,e$ satisfying the binomial inequality, the $q$-ary maximum $C_q(n,t,w,e)$ grows like $(q-1)^{w-e} \\binom{n}{w-e}/\\binom{w+t}{w-e}$, meaning the cooling, low-power, and error-correction requirements do not reduce the asymptotic rate below the packing or constant-weight bound.","The constant-power variant $C'_q(n,t,w,e)$ satisfies the same asymptotic formula, with exact equality available when generalized Steiner systems exist, such as for binary codes or for $e \\in \\{w-1,w-2\\}$ under divisibility conditions.","For small $w$, the paper proves $\\liminf_{n\\to\\infty} C(n,t,4,2)/n^2 \\ge 1/(12(t+1))$, and it conjectures that the small-$w$ regime has rate $1/((t+1)w(w-1))n^2$, distinct from the large-$w$ rate."],"supporting_citations":[{"why":"Introduces LPECC codes and the bus model with properties A(t), B(w), C(e) that the maximum-size problem is about.","marker":"[6]"},{"why":"Supplies the set-theoretic representation of LPECC codes, the packing construction $C(n,t,w,e) \\ge D(n+1,w+t,w-e)$, and previous exact values for small parameter cases that this paper extends.","marker":"[17]"},{"why":"Supplies near-optimal packings via the nibble method, which makes the $e=w-2$ upper bound asymptotically sharp.","marker":"[20]"},{"why":"Supplies the existence of balanced incomplete block designs under divisibility conditions, giving the exact equality and extremal design structure in Corollary 3.6 and Theorem 3.7.","marker":"[26, 27, 28]"},{"why":"Supplies the extremal graph bound used to prove the exact $(n,t,3,1)$ upper bound.","marker":"[24]"},{"why":"Supplies existence results for the 3-frames and 4-frames used in the small-$w$ lower-bound constructions.","marker":"[10]"},{"why":"Supplies near-optimal constant-weight codes of fixed odd distance, the lower-bound engine behind the $q$-ary asymptotic result.","marker":"[16]"},{"why":"Gives the Johnson-type upper bound for $q$-ary constant-weight codes used to control the low-weight parts in Theorem 5.1.","marker":"[18]"}],"fun_headline_variants":["Tight size bounds for low-power error-correcting bus codes","LPECC codes: exact maximum size found for large error budgets","Binomial ratio sets cap for power-saving bus codes","Error-correcting cooling codes: optimal size via block designs","Asymptotic exactness for low-power bus coding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The asymptotic lower bound rests on the unproved-in-this-paper existence of near-perfect constant-weight codes of exactly the weight and distance used in the construction; if that existence fails for some parameter choices, the claimed limit could be too high.","fun_headline_variants_meta":{"raw":{"variants":["Tight size bounds for low-power error-correcting bus codes","LPECC codes: exact maximum size found for large error budgets","Binomial ratio sets cap for power-saving bus codes","Error-correcting cooling codes: optimal size via block designs","Asymptotic exactness for low-power bus coding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000509,"raw_usage":{"total_tokens":2589,"prompt_tokens":1166,"completion_tokens":1423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":782,"completion_tokens_details":{"reasoning_tokens":1341}},"tokens_in":782,"tokens_out":1423,"duration_ms":10618,"temperature":1.0,"reasoning_tokens":1341,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:55:38.993296+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix $q=2$ and parameters $(t,w,e)$ satisfying $2\\binom{w}{w-e} \\ge \\binom{w-t-1}{w-e}+\\binom{w+t}{w-e}$, and compute $A_2(n,2(e+t)+1,w+t)$ along a sequence of $n$. If for some fixed $\\varepsilon>0$ infinitely many $n$ fall below $(1-\\varepsilon)\\binom{n}{w-e}/\\binom{w}{w-e}$, the construction's lower bound fails and $C_2(n,t,w,e)$ cannot reach the claimed asymptotic value.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces LPECC codes and the bus model with properties A(t), B(w), C(e) that the maximum-size problem is about."},{"cited_title":"Liu and L","cited_arxiv_id":null,"evidence_quote":"Supplies the set-theoretic representation of LPECC codes, the packing construction $C(n,t,w,e) \\ge D(n+1,w+t,w-e)$, and previous exact values for small parameter cases that this paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies near-optimal packings via the nibble method, which makes the $e=w-2$ upper bound asymptotically sharp."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the extremal graph bound used to prove the exact $(n,t,3,1)$ upper bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies existence results for the 3-frames and 4-frames used in the small-$w$ lower-bound constructions."},{"cited_title":"Liu and C","cited_arxiv_id":null,"evidence_quote":"Supplies near-optimal constant-weight codes of fixed odd distance, the lower-bound engine behind the $q$-ary asymptotic result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Johnson-type upper bound for $q$-ary constant-weight codes used to control the low-weight parts in Theorem 5.1."}],"review_version":1}