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In this paper the circulant Hadamard conjecture is proved., Comment: 23 pages

In this paper the relationship between $S$-subgroups in $\Z_{2}^{n}$ and binary codes is shown. If the codes used are both $P(T)$-codes and $G$-codes, then the $S$-subgroup is free. The codes constructed are cyclic, decimated or symmetric and the $S$-subgroups obtained are free under the action the cyclic permutation subgroup, invariants under the action the decimated permutation subgroup and symmetric under the action of symmetric permutation subgroup, respectively. Also it is shows that there is no codes generating whole $\Z_{2}^{n}$ in any $\G_{n}(a)$-complete $S$-set of the $S$-ring $\mathfrak{S}(\Z_{2}^{n},S_{n})$., Comment: 22 pages

In this paper three Schur ring are discussed, namenly: Hamming, circulant orbists and decimated circulant orbits Schur ring. By using autocorrelation function and the run structure of binary sequences we proof the relation between this Schur ring and combinatorial structures such as Hadamard matrices, periodic compatible binary sequences and perfect binary sequences. Cai proved for binary sequences that the autocorrelation function is in fact completely determined by its run structure. Also, he characterised the structure of the circulant Hadamard matrices. We characterise a more general structure, called periodic compatible binary sequences ($PComS$ for brevety), which generalises Hadamard matrices, periodic complementary binary sequences and binary sequences with $2$-level autocorrelation. Families of periodic compatibles binary sequences are presented. Also, we compute a bounds on familias $PComS$ in Hamming Schur ring. The results obtained are applied to families of $PComS$ such as circulant, with one and two circulant cores, Goethals-Seidel type and partial Hadamard matrices and perfect binary sequences., Comment: 23 pages

Turyn prove that if a circulant Hadamard matrix of order $n$ exists then $n$ must be of the form $n=4m^{2}$ for some odd integer $m$. In this paper we use the structure constant of Schur ring of $\Z_{2}^{4m^{2}}$ to prove that there is no circulant Hadamard matrix in $\Z_{2}^{4m^{2}}$ except possibly for sequences with Hamming weight $a+b$, with $\frac{m^{2}-m}{2}\leq a\leq\frac{3m^{2}-m}{2}$ and $b=2m^{2}-m-a$ and with $\frac{m^{2}+m}{2}\leq 2m^{2}-a\leq\frac{3m^{2}+m}{2}$ and $b=m+a$., Comment: 7 pages, substantial text overlap eliminated

In this paper a variety of issues are discussed, Schur ring, $S$-sets, circulant orbits, decimation operator and Hadamard matrices and their relation between them is shown. Firstly we define the complete $S$-sets. Next, we study the structure of Schur ring with circulant basic sets over $\Z_{2}^{n}$ and we define the free and non-free circulant $S$-sets, the symmetric, non-symmetric and antisymmetric circulant $S$-sets. We prove that all this $S$-sets are invariants under decimation. Finally, we prove that if a Hadamard matrix exist then this is contained in a complete $S$-set. Also, we prove that can't exist circulant and with one core Hadamard matrices with some particular structure. These theorems include a result known on symmetric circulant Hadamard matrices of order $4n$ only when $n$ is an odd number., Comment: 23 pages