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We give a survey of results and applications relating to the theory of Grobner bases of ideals and modules where the coefficient ring is a finite commutative ring. For applications, we specialize to the case of a finite chain ring. We discuss and compare the main algorithms that may be implemented to compute Grobner and (in the case of a chain ring) Szekeres-like bases. We give an account of a number of decoding algorithms for alternant codes over commutative finite chain rings.

We compute the Walsh spectrum of a new quadratic APN function, x3 +Tr(x9), showing that its Walsh transform is 3-valued for odd n, and is 5-valued for even n. Therefore, the distribution of the values of the Walsh transform of x3 + Tr(x9) is the same as that of the APN Gold functions. Moreover, for odd n the function is AB, which gives an alternative proof of the APN property of the function.

We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.