MinT - Best Known (238−85, 238, s)-Nets in Base 2

Best Known (238−85, 238, s)-Nets in Base 2

(238−85, 238, 112)-Net over F₂ — Constructive and digital

Digital (153, 238, 112)-net over F₂, using

2 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]

(238−85, 238, 138)-Net over F₂ — Digital

Digital (153, 238, 138)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−85, 238, 764)-Net in Base 2 — Upper bound on s

There is no (153, 238, 765)-net in base 2, because

1 times m-reduction [i] would yield (153, 237, 765)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 225238 256007 731534 638668 472608 348073 473992 642356 879000 751922 201555 017586 > 2²³⁷ [i]