MinT - Best Known (243−87, 243, s)-Nets in Base 2

Best Known (243−87, 243, s)-Nets in Base 2

(243−87, 243, 112)-Net over F₂ — Constructive and digital

Digital (156, 243, 112)-net over F₂, using

3 times m-reduction [i] based on digital (156, 246, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 123, 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]

(243−87, 243, 140)-Net over F₂ — Digital

Digital (156, 243, 140)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−87, 243, 773)-Net in Base 2 — Upper bound on s

There is no (156, 243, 774)-net in base 2, because

1 times m-reduction [i] would yield (156, 242, 774)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7 417755 839262 317610 109351 462987 933505 794334 339255 585652 005538 738927 437376 > 2²⁴² [i]