MinT - Best Known (242−77, 242, s)-Nets in Base 2

Best Known (242−77, 242, s)-Nets in Base 2

(242−77, 242, 112)-Net over F₂ — Constructive and digital

Digital (165, 242, 112)-net over F₂, using

t-expansion [i] based on digital (163, 242, 112)-net over F₂, using
- 18 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]

(242−77, 242, 183)-Net over F₂ — Digital

Digital (165, 242, 183)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−77, 242, 1163)-Net in Base 2 — Upper bound on s

There is no (165, 242, 1164)-net in base 2, because

1 times m-reduction [i] would yield (165, 241, 1164)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 581697 536475 206830 783683 455956 874044 324304 070754 627556 394386 226305 557512 > 2²⁴¹ [i]