MinT - Best Known (168, 168+75, s)-Nets in Base 2

Best Known (168, 168+75, s)-Nets in Base 2

(168, 168+75, 112)-Net over F₂ — Constructive and digital

Digital (168, 243, 112)-net over F₂, using

t-expansion [i] based on digital (163, 243, 112)-net over F₂, using
- 17 times m-reduction [i] based on digital (163, 260, 112)-net over F₂, using
  - trace code for nets [i] based on digital (33, 130, 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]

(168, 168+75, 197)-Net over F₂ — Digital

Digital (168, 243, 197)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+75, 1309)-Net in Base 2 — Upper bound on s

There is no (168, 243, 1310)-net in base 2, because

1 times m-reduction [i] would yield (168, 242, 1310)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7 070702 223999 881337 739634 776218 216292 912751 490450 467698 059107 352267 136872 > 2²⁴² [i]