MinT - Best Known (258−85, 258, s)-Nets in Base 2

Best Known (258−85, 258, s)-Nets in Base 2

(258−85, 258, 112)-Net over F₂ — Constructive and digital

Digital (173, 258, 112)-net over F₂, using

t-expansion [i] based on digital (163, 258, 112)-net over F₂, using
- 2 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]

(258−85, 258, 179)-Net over F₂ — Digital

Digital (173, 258, 179)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−85, 258, 1086)-Net in Base 2 — Upper bound on s

There is no (173, 258, 1087)-net in base 2, because

1 times m-reduction [i] would yield (173, 257, 1087)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 234234 515339 996158 379723 271261 386002 589954 757489 536964 667804 049009 726826 870285 > 2²⁵⁷ [i]