Given an m x m Hadamard matrix one can extract m2 symmetric designs on m -1 points each of which extends uniquely to a 3-design. Further, when m is a square, certain Hadamard matrices yield symmetric designs on m points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary association of linear codes with designs. This leads naturally to the notion, defined for any prime p , of p-equivalence for Hadamard

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... ices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty 24 x 24 matrices fall into only six 2-equivalence classes. In the 16x16 case, 2-equivalence is identical to the standard equivalence, but our results illuminate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian 2-group, why two of the matrices cannot be obtained from a symmetric design on 16 points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.