MinT - Best Known (213−65, 213, s)-Nets in Base 2

Best Known (213−65, 213, s)-Nets in Base 2

(213−65, 213, 112)-Net over F₂ — Constructive and digital

Digital (148, 213, 112)-net over F₂, using

17 times m-reduction [i] based on digital (148, 230, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 115, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(213−65, 213, 181)-Net over F₂ — Digital

Digital (148, 213, 181)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−65, 213, 1215)-Net in Base 2 — Upper bound on s

There is no (148, 213, 1216)-net in base 2, because

1 times m-reduction [i] would yield (148, 212, 1216)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6628 552656 097743 741977 071414 475876 546125 152770 308696 209014 346295 > 2²¹² [i]