{"id":"49262ff4-eb34-4fd9-a203-d24357b36fa2","arxiv_id":"2412.21002","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For transmit-sensor-selection ISAC, the identifiability-constrained codebook size is exactly C(Nt-2,Q-2) for a uniform linear array and 1 for a nonredundant array, with general upper and lower bounds between them.","lead":"This paper studies a dual-use wireless setup where the same antenna array must both send data and detect radar targets, and different antenna subsets act as communication symbols. It calculates how many such symbols are possible without losing the ability to identify the maximum number of targets, finding that regular evenly spaced layouts allow many symbols while maximally sparse layouts allow only one.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Formal Proposition 1 is correct, but the abstract and conclusions state the ULA codebook result without the Nt≤Nr+1 condition; for Nt=5, Nr=2, Q=3 the claimed sufficiency of the two edge sensors fails and the codebook size is 1, not C(3,1)=3.","rationale":"The reader's ACCEPT is reasonable for Proposition 1 as formally stated, and I do not find a flaw in the counting arguments or in the cited identifiability equivalence; the Vandermonde-spark reasoning supports Eq. (5). The load-bearing problem is that the abstract and conclusions restate the ULA result without the Nt≤Nr+1 hypothesis, and that hypothesis is exactly what makes the two-edge-sensor sufficiency true. The counterexample Nt=5, Nr=2, NΣ=6, Q=3 shows the unqualified statement is false: only one of the three edge-containing 3-subsets has a contiguous full sum co-array. This is not merely a typo-level issue, because the advertised 'large codebook' for ULAs fails in a natural parameter regime (more Tx than Rx antennas), and an implementer following the abstract would select codewords that do not guarantee maximal identifiability. Since the formal proposition is sound, the appropriate resolution is a qualification rather than a rejection, hence CONDITIONAL rather than REJECT or UNCHANGED.","tokens_in":8664,"tokens_out":29305,"duration_ms":309477,"concrete_test":"Enumerate all S⊆U5 with |S|=3 and S+U2=U6; confirm that only {0,2,4} is admissible, so |C*|=1 versus the unqualified C(3,1)=3. Then repeat for Nt=10, Nr=2, Q=6 with Dt=U10, Dr=U2: count the admissible edge-containing 6-subsets; if the count is below C(8,4), the abstract's unconditional 'only two sensors' claim fails in general. This is a short exhaustive check and does not depend on the cited theorem from [20].","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 1 (Eq. 10) is explicitly conditional on Nt ≤ Nr+1, and within that regime the proof is sound. However, the abstract and conclusions claim without qualification that for a ULA 'satisfying the identifiability constraint only requires including two specific sensors' and that the ULA achieves a codebook 'comparable to the unconstrained case.' This is false when the Tx ULA is longer than the Rx array. Take Dt=U5, Dr=U2, so NΣ=6, and Q=3 (admissible since L=ceil(6/2)=3). The edge sensors are 0 and 4. Among the three subsets containing {0,4}, only S={0,2,4} satisfies S+Dr=U6; S={0,1,4} gives {0,1,2,4,5}, missing 3, and S={0,3,4} gives {0,1,3,4,5}, missing 2. Hence |C*(3,5,2,6)|=1, not C(3,1)=3. So the advertised large-codebook property is not intrinsic to the ULA geometry; it requires the additional condition Nt≤Nr+1, which is omitted from the paper's headline claims. The formal theorem remains correct, but readers applying the abstract's statement to configurations with more Tx than Rx elements will obtain codewords that do not guarantee maximal identifiability.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies transmit sensor selection for integrated sensing and communications (ISAC) in a monostatic MIMO system. Each codeword corresponds to a Q-element subset S of the transmit array D_t, and the authors require that each S achieves the maximum number of identifiable targets, which they translate (via a cited equivalence) into the sum-set condition S + D_r = D_t + D_r = U_{N_Σ}. The central contribution is a set of bounds on the number of such subsets. Lemma 1 gives an upper bound of C(N_t-2, Q-2) by showing that the two extreme transmit sensors must be included. Proposition 1 gives exact values for two canonical geometries: for a ULA (with the additional condition N_t ≤ N_r + 1) the bound is tight, yielding a codebook of size C(N_t-2, Q-2); for a nonredundant array the only admissible codeword is the full transmit array. Proposition 2 provides a constructive lower bound of C(N_t-L, Q-L) when L = N_Σ/N_r is an integer. The authors conclude that redundancy is necessary for sensor-selection communication and that the ULA supports a large codebook.","tokens_in":8973,"tokens_out":13215,"duration_ms":121708,"significance":"If the results hold, this is a useful first analytical step toward understanding the trade-off between sensing identifiability and communication codebook size in ISAC. The problem formulation is novel, and the counting arguments in Lemma 1 and Propositions 1 and 2 are clean and, within their stated hypotheses, correct. The explicit characterization for the ULA and nonredundant arrays is a concrete contribution that can guide array geometry selection in practice. The paper also honestly identifies its scope: the bounds leave an 'uncertainty region' for intermediate values of N_Σ, and the tightness of the ULA result is conditional on N_t ≤ N_r + 1.","major_comments":[{"comment":"The paper's central identifiability guarantee — that a sensor-selection waveform achieves maximal identifiability if and only if S + D_r = D_t + D_r = U_{N_Σ} — is imported from an unpublished arXiv preprint [20]. Since the title and the entire problem formulation rest on this equivalence, the authors should either provide a proof (or at least a precise statement of the theorem with all necessary conditions on U, D_t, D_r) or clearly present it as an assumption. As written, a reader cannot verify the sensing interpretation that motivates the codebook definition, especially because the equivalence is 'if and only if' and the paper's counting results alone do not establish any sensing property.","section":"2.2"}],"minor_comments":[{"comment":"There are a few typos in the proof: 'the sum set is S + D_t = U_{N_Σ}' should be 'S + D_r = U_{N_Σ}', and the chain 'U_L = D_t + D_r ⊇ S + D_r ⊇ D_1 + D_r = U_L' should use U_{N_Σ} (or U_{N_r L}) on both ends, since D_1 + D_r = U_{N_r L} = U_{N_Σ}.","section":"Section 4.3 (Proof of Proposition 2)"},{"comment":"The caption of Fig. 2 says the examples illustrate 'Proposition 2,' but the text in Section 4.2 refers to 'this fact, revealed by the proof of Proposition 1, is illustrated in Fig. 2a.' Please align the citation so the reader knows which proposition each subfigure is demonstrating.","section":"Figure 2 caption and surrounding text"},{"comment":"The sentence 'Any positive tuple (Q, N_t, N_r, N_Σ) is admissible ... if (6) and (8) are satisfied' is stated without proof. For non-integer N_Σ/N_r the paper does not provide a construction, and the claim that such tuples are always realizable by some (D_t, D_r) is not obvious. Please add a reference or a brief construction for general N_Σ.","section":"Section 4 (admissibility discussion)"},{"comment":"The maximizer of C(N_t-2, Q-2) is written as Q = ⌊N_t/2 − 1⌉ + 2, but this notation is ambiguous; the standard statement is that the maximum is attained at Q = floor((N_t-2)/2) + 2 or Q = ceil((N_t-2)/2) + 2. Please state it clearly.","section":"Section 4.4"},{"comment":"In the sentence following Eq. (6), 'Tx subarray S ⊆ D_r' should be 'S ⊆ D_t', since S is a subset of the transmit array. Also, in the introduction, 'identifiablity' is misspelled.","section":"Section 2.2 and Eq. (6)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a short, conference-style manuscript. The technical results in Lemma 1 and Propositions 1 and 2 are correct under their stated hypotheses, and the counting check is solid. The main risk is the abstract's overstatement of the ULA result, which is genuinely false outside the N_t ≤ N_r + 1 regime; this must be fixed in revision. The second concern is the reliance on [20] for the identifiability equivalence; since [20] is an arXiv preprint from the same authors, the editor may wish to encourage a proof or a more explicit statement. These issues are fixable without changing the main conclusions, so I recommend minor revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Rajamäki and Pal give the first exact codebook-size formulas for sensor-selection ISAC under an identifiability constraint: for ULA with Nt≤Nr+1, |C*| = C(Nt-2, Q-2), and for nonredundant arrays, |C*|=1. The optimization problem is new, and the proofs are clean. Lemma 1 is a neat necessity argument (edge sensors must be included), and Proposition 1's ULA sufficiency is exactly explained by the overlap condition. Proposition 2's nested construction is a good lower-bound trick. I checked the counting; it holds.\n\nThe soft spot is the abstract and conclusions, not the formal statements. Proposition 1 is conditional on Nt≤Nr+1, and that condition is doing real work. The abstract says a ULA 'only requires including two specific sensors' and achieves a codebook 'comparable to the unconstrained case' with no qualification. That's false when the Tx array is longer than the Rx array. Example: Dt=U5, Dr=U2, Q=3. The two edge sensors are 0 and 4. Among the three subsets containing both, only {0,2,4} gives S+Dr=U6; {0,1,4} gives {0,1,2,4,5} and {0,3,4} gives {0,1,3,4,5}. So |C*|=1, not C(3,1)=3. The paper's own Proposition 1 technically avoids this, but a reader using the abstract on Nt>Nr+1 configurations will get non-guaranteed codewords. This must be fixed before publication, either by adding the condition to the headline claims or by repositioning the ULA result as 'when the Tx array is no longer than the Rx array.'\n\nOther weaknesses are minor. The paper is narrow—noiseless identifiability, 1D collinear arrays, equal-sized subarrays, no rate analysis—but it is upfront about that. The dependency on the cited equivalence from [20] is external but reasonable; it's consistent with the contiguous-sum-set literature. The concavity remark in Section 4.4 is loose but not load-bearing.\n\nWho is this for? People working on sparse-array ISAC and index modulation will find the characterization useful, even if the regime is restrictive. The formal results deserve a serious referee. I would accept for review and ask for the condition fix; I'd also ask the authors to state explicitly that [20]'s theorem is assumed, or to sketch its proof.","headline":"The counting results are correct, but the ULA codebook claim in the abstract and conclusions overreaches: the Nt≤Nr+1 condition is essential and omitting it gives wrong codebook sizes.","tokens_in":9501,"tokens_out":1962,"would_cite":true,"duration_ms":17839,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A uniform linear transmit array can support a sensor-selection ISAC codebook whose size matches the unconstrained binomial count, while a nonredundant array supports exactly one codeword, so redundancy is necessary for index-modulation…","keywords":["integrated sensing and communications","sensor selection","index modulation","sparse arrays","sum co-array","identifiability","uniform linear array","codebook design"],"falsifier":"Brute-force enumeration settles the counting claim: for a ULA with small $N_t$ and $N_r$, list every $Q$-sensor subarray, compute $S+D_r$, and count how many equal $U_{N_t+N_r-1}$; Proposition 1 predicts exactly $\\binom{N_t-2}{Q-2}$ such subarrays whenever $N_t\\le N_r+1$, and any mismatch would disprove it. For the nonredundant case, the same enumeration should show that no proper subset of the transmit array satisfies $S+D_r=D_t+D_r$, leaving only the full array.","tokens_in":8465,"feed_emoji":"📡","tokens_out":13640,"duration_ms":115115,"temperature":0.7,"pith_summary":"This paper asks how many different subsets of transmit sensors can double as information-bearing codewords in an integrated sensing and communications (ISAC) system without sacrificing the radar's ability to identify the maximum number of targets its array geometry allows. The answer is fully characterized at two geometric extremes. For a uniform linear array (ULA) whose receive array is large enough relative to the transmit array, every codeword that contains the two end sensors preserves the contiguous sum co-array, so the optimal codebook has exactly $\\binom{N_t-2}{Q-2}$ entries, comparable to the unconstrained sensor-selection codebook $\\binom{N_t}{Q}$. For a nonredundant array, which has the largest sum co-array and therefore the best identifiability, the only admissible codeword is the full transmit array, so no communication is possible through sensor selection alone. The paper concludes that array redundancy is necessary for this form of index-modulation communication, and it provides general upper and lower bounds on the codebook size for intermediate geometries.","feed_headline":"Uniform linear arrays get large codebooks; nonredundant arrays get one","feed_subtitle":"Under identifiability constraints, only redundant arrays can carry information through which transmit sensors are on.","key_machinery":"The load-bearing object is the sum co-array condition $S+D_r=D_t+D_r=U_{N_\\Sigma}$ from Eq. (5): a rank-$|S|$ sensor-selection waveform achieves maximal identifiability of $N_\\Sigma/2$ targets exactly when the selected subarray's sum set with the fixed receive array is contiguous and equals the full sum co-array, a fact the paper takes from a cited companion result. The counting proof then pivots on Lemma 1, which shows the two outermost transmit sensors must belong to every admissible subarray, giving the upper bound $\\binom{N_t-2}{Q-2}$. For the ULA, including both end sensors is also sufficient under $N_t\\le N_r+1$, so the upper bound is tight; for a nonredundant array, strict size counting forces $Q=N_t$ and leaves a single codeword. A constructive lower bound is obtained by embedding a dilated ULA of $L=N_\\Sigma/N_r$ sensors in every codeword, yielding $\\binom{N_t-L}{Q-L}$ guaranteed codewords.","core_discovery":"The central claim is Proposition 1: when the sum co-array has the ULA size $N_\\Sigma=N_t+N_r-1$ and $N_t\\le N_r+1$, any $Q$-sensor transmit subarray that includes the two extreme sensors has the same contiguous sum set as the full transmit-receive pair, and the identifiability-maximizing codebook therefore has exactly $\\binom{N_t-2}{Q-2}$ codewords. At the opposite extreme, a nonredundant array with $N_\\Sigma=N_tN_r$ forces $Q=N_t$, and exactly one codeword (all transmit sensors active) is admissible. These two results bracket the trade-off: redundancy in the array geometry buys sensor-selection communication capability, while a large sum co-array buys target identifiability but consumes the degrees of freedom needed to encode information.","pith_inferences":["If the Eq. (5) equivalence holds for all contiguous sum co-arrays, the edge-sensor upper bound implies that no array geometry can exceed $\\binom{N_t-2}{Q-2}$ identifiability-maximizing codewords, making array redundancy a quantitative communications resource rather than only a sensing design choice.","The exact optimal codebook size for intermediate redundancy, $N_t+N_r-1