Locally decodable codes and the failure of cotype for projective tensor products

@article{Brit2012LocallyDC,
  title={Locally decodable codes and the failure of cotype for projective tensor products},
  author={J. Bri{\"e}t and A. Naor and O. Regev},
  journal={ArXiv},
  year={2012},
  volume={abs/1208.0539}
}
  • J. Briët, A. Naor, O. Regev
  • Published 2012
  • Mathematics, Computer Science
  • ArXiv
  • It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $l_p\hat\otimes X$ fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then $l_{p_1}\hat\otimes l_{p_2} \hat\otimes l_{p_3}$ does not have finite cotype. This is proved via a connection to the theory of locally decodable codes. 

    Topics from this paper.

    On Embeddings of l1k from Locally Decodable Codes
    2
    Snowflake universality of Wasserstein spaces
    18
    Arithmetic expanders and deviation bounds for random tensors
    2
    Lower Bounds for Approximate LDC
    1
    Convex Bodies Associated to Tensor Norms
    2
    Lower Bounds for Approximate LDCs
    5

    References

    Publications referenced by this paper.
    SHOWING 1-10 OF 44 REFERENCES
    Random Series of Trace Class Operators
    3
    Observations About the Projective Tensor Product of Banach Spaces, II — Lp(0, 1) ⊗X, 1 < p < ∞
    15
    Exponential lower bound for 2-query locally decodable codes via a quantum argument
    118
    3-query locally decodable codes of subexponential length
    184
    Superpolynomial Size Set-systems with Restricted Intersections mod 6 and Explicit Ramsey Graphs
    109
    Duals of tensor products
    14