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We initiate the study of property testing problems concerning equations in permutations. In such problems, the input consists of permutations σ 1 , . . . , σ d ∈ Sym(n), and one wishes to determine whether they satisfy a certain system of equations E, or are far from doing so. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that E is testable. For example, when d = 2 and E consists of the single equation XY = YX, this corresponds to testing whetherWe formulate the well-studied group-theoretic notion of stability in permutations as a testability concept, and interpret all works on stability as testability results. Furthermore, we establish a close connection between testability and group theory, and harness the power of group-theoretic notions such as amenability and property (T) to produce a large family of testable equations, beyond those afforded by the study of stability, and a large family of non-testable equations.Finally, we provide a survey of results on stability from a computational perspective and describe many directions for future research.

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We investigate the weight distribution of random binary linear codes. For 0 < λ < 1 and n→∞ pick uniformly at random λn vectors in double-struckF2n and let C≤double-struckF2n be the orthogonal complement of their span. Given 0 < γ < 1/2 with 0 < λ < h(γ) let X be the random variable that counts the number of words in C of Hamming weight γn. In this paper we determine the asymptotics of the moments of X of all orders ofalse(nnormallognfalse).