MinT - Best Known (239−71, 239, s)-Nets in Base 2

Best Known (239−71, 239, s)-Nets in Base 2

(239−71, 239, 112)-Net over F₂ — Constructive and digital

Digital (168, 239, 112)-net over F₂, using

t-expansion [i] based on digital (163, 239, 112)-net over F₂, using
- 21 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]

(239−71, 239, 212)-Net over F₂ — Digital

Digital (168, 239, 212)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−71, 239, 1498)-Net in Base 2 — Upper bound on s

There is no (168, 239, 1499)-net in base 2, because

1 times m-reduction [i] would yield (168, 238, 1499)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 446157 005815 221343 700772 024031 221545 805241 311237 095942 875081 515346 635876 > 2²³⁸ [i]