Juntas in theℓ₁-grid and Lipschitz maps between discrete tori

Abstract

We show that if A ⊂ [k]n , then A is -close to a junta depending upon at most exp(O(|∂A|/(k n−1 ))) coordinates, where ∂A denotes the edge-boundary of A in the 1 -grid. This bound is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [6], or as a characterisation of large subsets of the 1 -grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [1]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.

Input-Output Distance Properties of Good Linear Codes

“…The tradeoff stems from the fact that smoothness requires local structures (c.f. [12]), and these in turn cannot spread messages too far out.…”

“…Indeed, a code maybe optimal in the sense of minimum distance, yet, there may exists a larger code that achieves the same minimum distance. For instance, Tanner [14] constructed a binary [12,4,6]-code. The linear programming (LP) bound rules out the existence of a [12,3,7]-code.…”

“…For instance, Tanner [14] constructed a binary [12,4,6]-code. The linear programming (LP) bound rules out the existence of a [12,3,7]-code. Thus any [12,3,6]-subcode of the Tanner code is still optimal in the sense of minimum distance.…”

“…The linear programming (LP) bound rules out the existence of a [12,3,7]-code. Thus any [12,3,6]-subcode of the Tanner code is still optimal in the sense of minimum distance. In general, one can expect such codes to exists over any field where the singleton bound is not tight.…”

“…We ask if there exists a [14,4]-code that dominates the quasi-cyclic code of (12). The LP in (9) is infeasible if we set A * 2 = 7 while keeping the rest of A * i 's from above unchanged.…”