MinT - Best Known (68, 68+41, s)-Nets in Base 2

Best Known (68, 68+41, s)-Nets in Base 2

(68, 68+41, 60)-Net over F₂ — Constructive and digital

Digital (68, 109, 60)-net over F₂, using

1 times m-reduction [i] based on digital (68, 110, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 55, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(68, 68+41, 66)-Net over F₂ — Digital

Digital (68, 109, 66)-net over F₂, using

1 times m-reduction [i] based on digital (68, 110, 66)-net over F₂, using
- trace code for nets [i] based on digital (13, 55, 33)-net over F₄, using
  - net from sequence [i] based on digital (13, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
      - Drinfeld modules of rank 1 [i]

(68, 68+41, 322)-Net in Base 2 — Upper bound on s

There is no (68, 109, 323)-net in base 2, because

1 times m-reduction [i] would yield (68, 108, 323)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 339 211203 045411 721437 645862 503536 > 2¹⁰⁸ [i]