MinT - Best Known (238−81, 238, s)-Nets in Base 2

Best Known (238−81, 238, s)-Nets in Base 2

(238−81, 238, 112)-Net over F₂ — Constructive and digital

Digital (157, 238, 112)-net over F₂, using

10 times m-reduction [i] based on digital (157, 248, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 124, 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]

(238−81, 238, 154)-Net over F₂ — Digital

Digital (157, 238, 154)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−81, 238, 900)-Net in Base 2 — Upper bound on s

There is no (157, 238, 901)-net in base 2, because

1 times m-reduction [i] would yield (157, 237, 901)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 228370 499116 365193 501602 910071 534639 422776 935147 691738 104872 727216 341424 > 2²³⁷ [i]