One fundamental and challenging problem in coding theory is to optimize the parameters [n, k,d] of a linear code over the finite field Fq and construct codes with best possible parameters. There are tables and databases of best-known linear codes over the finite fields of size up to 9 together with upper bounds on the minimum distances. Motivated by recent works on codes over F₁₁, we present a table of best-known linear codes over F₁₁ together with upper bounds on minimum distances. Our table covers the range n ≤ 150 for the length, and 3 ≤ k ≤ 7 for the dimension. To the best of our knowledge, this is the first time such a table is presented in the literature. For the construction of the best-known codes, we employed an iterative heuristic search algorithm to search for new linear codes in the class of quasi-twisted (QT) codes. The search yielded many new codes with better parameters than previously known codes. In many cases, optimal codes are obtained. In addition to presenting a comprehensive table of best-known codes over F₁₁ of dimensions up to 7 with upper bounds on the minimum distances, we also present separate tables for the optimal codes and new QT codes over F₁₁. We hope that this work will be a useful source for further study on codes over F₁₁.