arXiv subjects

Chengu Wang

Publications and source records attributed to Chengu Wang.

6 recordsLinked to original sources

arXiv2026

A Lower Bound of 21 for $3\times3$ Matrix Multiplication over $\mathbb{F}_2$

We prove that $3\times3$ matrix multiplication over $\mathbb{F}_2$ has bilinear complexity at least $21$, improving the lower bound $20$. Lower bounds for restrictions of the first input constrain how many first factors of a decomposition can lie in each subspace. The strengthened restriction bounds force all first factors of matrix rank at least two into one coset of a three-dimensional rank-one subspace. An exhaustive computation finds no admissible first-factor profile with $20$ terms. We also prove that $2\times3$ by $3\times3$ matrix multiplication over $\mathbb{F}_3$ has rank exactly $15$: three profiles survive the corresponding computation, and short restriction arguments exclude them. We combine Wang's automated framework for tensor-rank lower bounds with D'Ambrosio's capacity-and-profile strategy. Our computational contributions are rank-one-span searches that strengthen the subspace lower-bound table and a direct, symmetry-reduced profile enumerator that enforces all subspace capacities simultaneously.

cs.CC↗

arXiv2026

Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43

We improve the best known sphere packings in every dimension from $38$ to $43$. Our packings in dimensions $38$ to $42$ come from one chain of cross-sections of the extremal even unimodular lattice $P_{48p}$, $\sqrt3E_6 \subset \sqrt3E_7 \subset \sqrt3E_8 \subset K_9 \subset K_{10}$, whose first three members are cut from the fixed lattice $\sqrt3(E_8 \perp E_8)$ of an order-three automorphism; their orthogonal complements are lattice packings and set the records in dimensions $42$ down to $38$. Each successive section is a determinant-minimal extension of its predecessor. Our $43$-dimensional packing is an antipode packing: six translates of the complement of a $5$-dimensional section. We also improve some kissing numbers. Conway and Sloane's twelve $1982$ cross-section packings appear never to have had theirs computed; we compute them and find that, in dimensions $42$ to $47$, they exceed the previously tabulated lower bounds. Our chain does better in dimensions $40$, $41$ and $42$, and a further antipode packing beats the record in dimension $45$.

math.MG↗

arXiv2026

Automated Lower Bounds for Bilinear Complexity over Finite Fields

We present a general, automated framework for proving lower bounds on the bilinear complexity (tensor rank) of multiplication problems over a finite field $\mathbb{F}_q$. The framework is parameterized only by the multiplication tensor and by a group of rank-preserving symmetries acting on one argument: it classifies the subspaces of the argument into orbits under the group, runs a dynamic program over the orbits combining four lower-bound techniques, and emits a proof certificate that a verifier rechecks, typically faster than the search. Instantiating the framework for matrix multiplication, we improve the lower bounds for three small formats over $\mathbb{F}_2$, most notably showing that the bilinear complexity of multiplying two $3 \times 3$ matrices over $\mathbb{F}_2$ is at least $20$, raising the bound of $19$ that had stood since Bl\"aser (2003). Instantiating it for polynomial multiplication, we obtain eighteen new lower bounds over $\mathbb{F}_2$ and $\mathbb{F}_3$, for the full product, cyclic convolution, and the truncated (modulo $x^N$) and negacyclic (modulo $x^N+1$) products. Every bound is backed by a machine-checkable certificate.

cs.CC↗

arXiv2025

Depth-13 Sorting Networks for 28 Channels

We establish new depth upper bounds for sorting networks on 27 and 28 channels, improving the previous best bound of 14 to 13. Our 28-channel network is constructed with reflectional symmetry by combining high-quality prefixes of 16- and 12-channel networks, extending them greedily one comparator at a time, and using a SAT solver to complete the remaining layers.

cs.DS↗

arXiv2014

On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model

We study the communication complexity of linear algebraic problems over finite fields in the multi-player message passing model, proving a number of tight lower bounds. Specifically, for a matrix which is distributed among a number of players, we consider the problem of determining its rank, of computing entries in its inverse, and of solving linear equations. We also consider related problems such as computing the generalized inner product of vectors held on different servers. We give a general framework for reducing these multi-player problems to their two-player counterparts, showing that the randomized $s$-player communication complexity of these problems is at least $s$ times the randomized two-player communication complexity. Provided the problem has a certain amount of algebraic symmetry, which we formally define, we can show the hardest input distribution is a symmetric distribution, and therefore apply a recent multi-player lower bound technique of Phillips et al. Further, we give new two-player lower bounds for a number of these problems. In particular, our optimal lower bound for the two-player version of the matrix rank problem resolves an open question of Sun and Wang. A common feature of our lower bounds is that they apply even to the special "threshold promise" versions of these problems, wherein the underlying quantity, e.g., rank, is promised to be one of just two values, one on each side of some critical threshold. These kinds of promise problems are commonplace in the literature on data streaming as sources of hardness for reductions giving space lower bounds.

cs.CC↗

arXiv2013

The Garden Hose Complexity for the Equality Function

The garden hose complexity is a new communication complexity introduced by H. Buhrman, S. Fehr, C. Schaffner and F. Speelman [BFSS13] to analyze position-based cryptography protocols in the quantum setting. We focus on the garden hose complexity of the equality function, and improve on the bounds of O. Margalit and A. Matsliah[MM12] with the help of a new approach and of our handmade simulated annealing based solver. We have also found beautiful symmetries of the solutions that have lead us to develop the notion of garden hose permutation groups. Then, exploiting this new concept, we get even further, although several interesting open problems remain.

quant-ph↗