MinT - Best Known (216−72, 216, s)-Nets in Base 2

Best Known (216−72, 216, s)-Nets in Base 2

(216−72, 216, 112)-Net over F₂ — Constructive and digital

Digital (144, 216, 112)-net over F₂, using

6 times m-reduction [i] based on digital (144, 222, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 111, 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]

(216−72, 216, 149)-Net over F₂ — Digital

Digital (144, 216, 149)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−72, 216, 861)-Net in Base 2 — Upper bound on s

There is no (144, 216, 862)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 106029 421137 994052 277098 366725 244138 617889 080085 769364 828104 666894 > 2²¹⁶ [i]