Construction of Hadamard codes using Hadamard Rhotrice's essay
Nebe and Villar gave a series of ternary self-dual codes of length, 2 p 1, 2, p, 1, for a prime number p congruent. As a result, the third ternary extremal self-dual code was found. We show that these ternary self-dual codes contain code words that form a Hadamard matrix of order, 2 p 1, 2, p. This article narrows this gap and shows the existence of codes for requests with servers. Another construction in the paper creates a code with servers and requests, which is an optimal result. We provide some limits on the minimum number of servers for functional batch codes. These constructs are mainly based on Hadamard, the Hadamard transform and error correction coding, a naturalness-preserving transform for signaling and other constructs of Hn. Preface. 1. Kronecker matrix algebra. 2. A Hadamard matrix. 3. The Sylvester-Hadamard matrix of n. 4. The eigenvalues of Hn. 5. The eigenvectors of Hn. 6. Other constructions of Hn. 7; In we will consider the unimodular lattices obtained from the Z m code and the Z n, code of a binary Hadamard matrix of order n. In particular, it is shown that the lattice obtained from the ternary code of a Hadamard matrix H of is isometric with respect to a neighbor L of the lattice obtained from the binary code of H. Hadamard matrix H by substituting each entry gij by the nn-permutation matrix R.gij given by Cayley's representation. The matrices B and C are rectangular in size. nC1 n. n C1 n C1 respectively, and are not dependent on the geometry. Since C is symmetric, A is symmetric. 0 1 -matrix if and only ifR.H is symmetric and follows from this. The proposed concatenated ZH codes offer comparable performance to another class of low-rate codes, the turbo Hadamard codes, and better performance than superorthogonal turbo codes, with much lower coding and decoding complexities.