It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length \(2q+2\) contains the (0,1)-incidence matrix of a Hadamard 3-\((2q+2,q+1,(q-1)/2)\) design D(q) associated with a Paley–Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley–Hadamard matrix of type I. If \(q=5, 11, 17, 23\), then the full permutation automorphism group of L(q) coincides with the full automorphism group of D(q), and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code C(17) are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley–Hadamard matrix H of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix \(H'\) such that the symmetric 2-(36, 15, 6) design D associated with \(H'\) has trivial full automorphism group, and the incidence matrix of D spans a ternary code equivalent to C(17).

证明了单项式等价于长度为\(2q+2\)的 Pless 对称码C ( q ) 的码L ( q )包含 Hadamard 3- \(( 2q+2,q+1,(q-1)/2)\)设计D ( q ) 与类型 II 的 Paley-Hadamard 矩阵相关联。类似地，任何三元扩展二次残差代码都包含与类型 I 的 Paley-Hadamard 矩阵相关联的 Hadamard 3-设计的关联矩阵。 如果\(q=5, 11, 17, 23\)，则完全置换自同构群的大号（q）与全同构组的一致d（q )，并且类似的结果适用于长度为 24 和 48 的三元扩展二次残差码。由 Pless 对称码C (17)的码字形成的所有 36 阶 Hadamard 矩阵被枚举并分类为等价。这种矩阵有两个等价类：I类的 Paley-Hadamard 矩阵H，具有阶为 19584 的完全自同构群，以及第二个正则 Hadamard 矩阵\(H'\)使得对称 2-(36, 15, 6）设计d相关联\（H'\）具有微不足道的全自同构组，和的关联矩阵d跨越一三元代码相当于ç（17）。