MinT - Best Known (239−79, 239, s)-Nets in Base 2

Best Known (239−79, 239, s)-Nets in Base 2

(239−79, 239, 112)-Net over F₂ — Constructive and digital

Digital (160, 239, 112)-net over F₂, using

15 times m-reduction [i] based on digital (160, 254, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 127, 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]

(239−79, 239, 166)-Net over F₂ — Digital

Digital (160, 239, 166)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−79, 239, 1001)-Net in Base 2 — Upper bound on s

There is no (160, 239, 1002)-net in base 2, because

1 times m-reduction [i] would yield (160, 238, 1002)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 451589 026056 591812 975659 404074 003724 094786 118415 536565 021187 390668 532284 > 2²³⁸ [i]