MinT - Best Known (232−81, 232, s)-Nets in Base 2

Best Known (232−81, 232, s)-Nets in Base 2

(232−81, 232, 112)-Net over F₂ — Constructive and digital

Digital (151, 232, 112)-net over F₂, using

4 times m-reduction [i] based on digital (151, 236, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 118, 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]

(232−81, 232, 142)-Net over F₂ — Digital

Digital (151, 232, 142)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−81, 232, 805)-Net in Base 2 — Upper bound on s

There is no (151, 232, 806)-net in base 2, because

1 times m-reduction [i] would yield (151, 231, 806)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3488 908386 998543 950226 776889 553156 200586 764340 299092 804716 543968 330880 > 2²³¹ [i]