MinT - Best Known (223−77, 223, s)-Nets in Base 2

Best Known (223−77, 223, s)-Nets in Base 2

(223−77, 223, 112)-Net over F₂ — Constructive and digital

Digital (146, 223, 112)-net over F₂, using

3 times m-reduction [i] based on digital (146, 226, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 113, 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]

(223−77, 223, 141)-Net over F₂ — Digital

Digital (146, 223, 141)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−77, 223, 807)-Net in Base 2 — Upper bound on s

There is no (146, 223, 808)-net in base 2, because

1 times m-reduction [i] would yield (146, 222, 808)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7 035594 724986 772590 356950 875129 002484 490840 217555 249420 925593 639071 > 2²²² [i]