MinT - Best Known (258−81, 258, s)-Nets in Base 2

Best Known (258−81, 258, s)-Nets in Base 2

(258−81, 258, 112)-Net over F₂ — Constructive and digital

Digital (177, 258, 112)-net over F₂, using

t-expansion [i] based on digital (163, 258, 112)-net over F₂, using
- 2 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]

(258−81, 258, 199)-Net over F₂ — Digital

Digital (177, 258, 199)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−81, 258, 1296)-Net in Base 2 — Upper bound on s

There is no (177, 258, 1297)-net in base 2, because

1 times m-reduction [i] would yield (177, 257, 1297)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 234512 633707 506343 865902 593644 168559 652826 848856 244675 062577 333511 559952 471568 > 2²⁵⁷ [i]