{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace random or learned routing hashes for tokens arranged on a d by d grid with a fixed family of mutually orthogonal anti-Latin squares. Each channel assigns exactly d of the d squared tokens to every bucket, while any two channels jointly distinguish every grid position. The resulting router has deterministic load balance and multi-view positional diversity.",
 "formulas_latex": [
  "$$\\left|\\{(i,j)\\in[d]^2:A_{i,j}=q\\}\\right|=d\\qquad\\text{for every symbol }q\\in[d].$$",
  "$$\\forall r\\ne s:\\quad (i,j)\\mapsto\\bigl(A^{(r)}_{i,j},A^{(s)}_{i,j}\\bigr)\\text{ is one-to-one}.$$",
  "$$\\left|\\{(i,j):(A^{(r)}_{i,j},A^{(s)}_{i,j})=(u,v)\\}\\right|=1\\qquad\\text{for every }(u,v)\\in[d]^2.$$$$",
  "$$N_L(d)+1\\le N_A(d)\\le N_L(d)+2,\\qquad N_A(d)=N_L(d)+2\\text{ for }d\\ge4.$$"
 ],
 "id": 2865,
 "implementation": "Integrate the construction at the routing decision in an MoE layer whose tokens correspond to a d by d image-patch grid or a sequence reshaped into that grid. Prepare K mutually orthogonal anti-Latin squares offline and store them as an integer tensor A with shape [K,d,d]. For token position (i,j), select a channel r and compute q = A[r,i,j]. In a fixed top-1 router, map q directly to expert q when the number of experts is d. For two-channel routing, compute q1 = A[r,i,j] and q2 = A[s,i,j], then use the pair as a structured code; with d squared experts use expert index q1 times d plus q2, while with fewer experts hash the pair but retain both labels as auxiliary features. Rotate r and s across layers or batches. A practical hybrid is score_e = alpha times fixed_score_e plus (1-alpha) times learned_gate_e(x), with alpha initially near one and decayed if learned routing improves accuracy. The exact load count is computed from the identity sum over i,j of indicator[A[r,i,j] = q] = d, so no stochastic estimate or load-balancing loss is needed for the fixed component. Measure expert counts, overflow, token dropping, pairwise co-routing collisions, router entropy, and training variance. First test a Switch-style Transformer or ViT-Tiny on CIFAR-100 with d = 8, 64 tokens, and 8 experts. Compare anti-Latin routing against random balanced hashing and learned top-1 routing with an auxiliary load loss at equal FLOPs. Success is zero or near-zero overflow, lower batch-to-batch expert-load variance, and equal or better validation accuracy without the auxiliary balancing term.",
 "math_summary": "An anti-Latin square A of order d is a d by d matrix with entries in the alphabet [d] = {0, ..., d - 1}, such that every symbol occurs exactly d times in the whole matrix and at least one symbol repeats in every row and every column. Two anti-Latin squares A^(r) and A^(s) are orthogonal when the map from cell position (i,j) to the ordered pair (A^(r)_(i,j), A^(s)_(i,j)) is one-to-one. Since both the set of cells and the set of ordered pairs have size d squared, every ordered pair appears exactly once. The paper proves N_L(d) + 1 \u003c= N_A(d) \u003c= N_L(d) + 2, and N_A(d) = N_L(d) + 2 for d \u003e= 4, where N_A and N_L are the maximum sizes of mutually orthogonal anti-Latin and Latin-square families. The transferable facts are exact marginal balance, pairwise collision-freedom, and the existence of several deterministic channels.",
 "math_tags": [
  "combinatorics",
  "information-theory",
  "finite-geometry"
 ],
 "ml_areas": [
  "moe-routing",
  "attention",
  "transformer"
 ],
 "paper": {
  "arxiv_id": "2608.30082",
  "arxiv_url": "https://arxiv.org/abs/2608.30082",
  "summary_what_math_gives_to_ml": "The paper gives deterministic families of categorical grids with three useful properties: every channel is globally balanced, every row and column contains a repeated symbol, and every pair of channels uniquely identifies a grid cell. This structure can replace random or independently learned hash assignments in MoE routing or two-dimensional positional coding. The most practical transfer is a fixed multi-channel routing scheme that guarantees exact marginal bucket balance and pairwise diversity without a learned routing table or an auxiliary load-balancing objective.",
  "title": "Mutually orthogonal anti-Latin squares",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 5
 },
 "solves": [
  "scalability",
  "memory",
  "stability"
 ],
 "title": "Orthogonal anti-Latin hash routing",
 "url": "https://synthcore.org/idea/2865/orthogonal-anti-latin-hash-routing",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}