Lower Bounds on Formula Size of Error-Correcting Codes ⋆ Arist Kojevnikov and Alexander S. Kulikov St. Petersburg Department of Steklov Institute of Mathematics 27 Fontanka, 191023 St.Petersburg, Russia http://logic.pdmi.ras.ru/{~arist,~kulikov}/ Abstract. We show that every formula over the basis {∧, ∨, ¬} for a function f : {0, 1} n →{0, 1}, such that ∀x, y ∈ f -1 (1),d(x, y) ≥ 2d + 1, has size Ω(n d+2 |f -1 (1)| |f -1 (0)| ) . This immediately implies a lower bound Ω(n 2 ) for a characteristic func- tion of a BCH code of distance 2d + 1. The main technique used is estimating the number of monochromatic rectangles needed to cover a matrix. 1 Introduction One of the most important problems in theoretical computer science is proving lower bounds for various computational models. Boolean circuits is probably the simplest such model. A Boolean circuit has n inputs, one output, interior gates that are labeled by Boolean functions (usually, these are ∧, ∨ and ¬) and computes in a natural way a function f : {0, 1} n →{0, 1}. By general counting arguments it is possible to show that almost every Boolean function has ex- ponential circuit complexity. Despite of this no nonlinear lower bound on the circuit size of an explicit Boolean function is known. Some progress however has been made in restricted settings. Razborov [2] proved a superpolynomial lower bound on the monotone com- plexity of the clique function. Exponential lower bounds are also known for constant depth circuits (see, e.g., [3] and references therein). In this paper we consider another restricted case of Boolean circuits, namely Boolean formulas. A formula is just a circuit whose underlying graph is a tree. While a formula is weaker than a circuit, it is known [4] that for any Boolean function the minimal depth D(f ) of a circuit and the logarithm of the minimal size L(f ) of a formula computing the same function f have the same asymptotic behavior, i.e., ∀f : {0, 1} n →{0, 1}, D(f )= Θ(log L(f )) . ⋆ After we finished this paper, we were told that the result was proved (by more simple method) in [1]. However, you may still want to read our draft to get the details of the proof (if, for example, you do not know Russian).