communication protocol designers

Abstract: Local coordinate-wise linear (LCL) properties provide a unified framework for studying list decoding, list recovery and other local properties of linear codes. We prove that random one-point algebraic geometry (AG) codes have the same LCL threshold rates as random linear codes under explicit alphabet and sampling conditions. More precisely, for a family of local profiles with random linear code threshold $R_{\mathcal P}$, a random one-point AG code avoids all such profiles below $R_{\mathcal P}-\varepsilon$ and contains one above $R_{\mathcal P}+\varepsilon$, with explicit failure bounds on both sides. Our proof also removes an auxiliary lower bound on the number of available rational places that arises in a direct adaptation of the Reed--Solomon argument and requires no additional genus-dependent rate gap. We also observe that the previously claimed above threshold proof for random Reed--Solomon codes does not establish the required probability lower bound. The containment relation used in that argument gives the probability comparison in the opposite direction from what is needed. We give a different above threshold argument, which in particular also yields an alternative proof in the genus zero case. Combining our threshold theorem with recent results for random linear codes gives a collection of new list decoding, average weight list decoding and list recovery guarantees for random one-point AG codes. We further obtain new correlated agreement and proximity gap results for polynomial curves. These results extend the LCL threshold framework from random linear and Reed--Solomon codes to random one-point AG codes.

Abstract: We develop a multi-view linear-programming framework for locally recoverable codes with arbitrary fixed availability $a$. For any retained order $1 \le s \le a$, the selected helper sets, the recovered coordinate, and their complement form an $(s+2)$-part partition. Recording the Hamming distance on all blocks preserves both compatibility among the selected repair alternatives and their coupling with the remaining coordinates. The resulting joint distribution satisfies centered counting identities, product-Krawtchouk positivity, and collision inequalities from the local-distance condition; for fixed $s$, these constraints give a polynomial-size relaxation for arbitrary, possibly nonlinear, codes over any finite field. Retaining one view recovers the three-block model of our companion paper. We develop the first genuinely multi-view case, $s=2$, in detail and specialize the exact computations to availability $a=2$. The resulting four-block LP projects to both the one-view three-block model and a globally conditioned two-view relaxation, making explicit the information lost by each coarsening. Exact rational primal-dual certificates together with checked constructions prove $M_{\max}(2,8,4,2,2,2)=8$, $M_{\max}(3,7,3,2,2,2)=27$, and $M_{\max}(4,7,3,2,2,2)=64$; in all three cases the four-block bound is strictly stronger than both coarsenings.

Abstract: A communication problem can have far more possible inputs than its communication cost would suggest. Must its difficulty already be present on a much smaller set of inputs? We prove that every finite total Boolean matrix of deterministic communication complexity $c\ge4$ has a submatrix on $2^k$ of its original rows and $2^k$ of its original columns, with $k=Θ_\varepsilon(c)$ and complexity at least $(1-\varepsilon)(k+1)$, for every fixed $0<\varepsilon<1$. Since $k+1$ is the maximum possible cost on such a square, the retained problem can be arbitrarily close to maximally hard. This answers affirmatively the lossless condensation question of Hamed Hatami; Göös, Newman, Riazanov, and Sokolov (STOC 2024), who recorded it as Open Problem 2, conjectured a negative answer. Hrubeš previously guaranteed input length $Ω(\sqrt c)$. The same argument gives an original $2^{c-2}$-by-$2^{c-2}$ square retaining at least $c/3-O(\log c)$ bits of communication complexity. The proof builds on Hrubeš's counting and covering argument. We count submatrices equipped with short communication protocols: a player names a covering submatrix, then the players run its protocol. This avoids the loss from converting rectangle partitions into protocols. A recursion on rectangles makes the argument constructive. For fixed rational $\varepsilon$ and any target depth $d\ge4$, a deterministic algorithm returns either a protocol of depth below $d$, or a square of original inputs at input length $Θ_\varepsilon(d)$ with the same near-maximal guarantee. Its running time is $2^{O(2^d)}$ times a polynomial in the table size. If the original complexity is at least $d$, the algorithm necessarily returns the square. An extension gives constant-factor condensation for any fixed number of number-in-hand players, with bounds independent of the finite output alphabet.