MinT - Best Known (222−69, 222, s)-Nets in Base 2

Best Known (222−69, 222, s)-Nets in Base 2

(222−69, 222, 112)-Net over F₂ — Constructive and digital

Digital (153, 222, 112)-net over F₂, using

18 times m-reduction [i] based on digital (153, 240, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 120, 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]

(222−69, 222, 179)-Net over F₂ — Digital

Digital (153, 222, 179)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−69, 222, 1175)-Net in Base 2 — Upper bound on s

There is no (153, 222, 1176)-net in base 2, because

1 times m-reduction [i] would yield (153, 221, 1176)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 414363 044661 109532 334901 152974 211215 152716 250360 222683 673695 064548 > 2²²¹ [i]