p-ADIC WELCH BOUNDS AND p-ADIC ZAUNER CONJECTURE

- K. Mahesh Krishna

## Abstract

Let p be a prime. For d ∈ N, let Q d p be the standard d-dimensional p-adic Hilbert space. Let m ∈ N and Sym m (Q d p) be the p-adic Hilbert space of symmetric m-tensors. We prove the following result. Let {τ j } n j=1 be a collection in Q d p satisfying (i) τ j , τ j = 1 for all 1 ≤ j ≤ n and (ii) there exists b ∈ Q p satisfying [ n j=1 x, τ j τ j = bx for all x ∈ Q d p. Then max 1≤j,k≤n,j =k {|n|, ||τ j , τ k | 2m } ≥ |n| 2 d+m−1 m. (1) We call Inequality (1) as the p-adic version of Welch bounds obtained by Welch [IEEE Transactions on Information Theory, 1974 ]. Inequality (1) differs from the non-Archimedean Welch bound obtained recently by M. Krishna as one can not derive one from another. We formulate p-adic Zauner conjecture.