MinT - Best Known (258−87, 258, s)-Nets in Base 2

Best Known (258−87, 258, s)-Nets in Base 2

(258−87, 258, 112)-Net over F₂ — Constructive and digital

Digital (171, 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]

(258−87, 258, 169)-Net over F₂ — Digital

Digital (171, 258, 169)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−87, 258, 1000)-Net in Base 2 — Upper bound on s

There is no (171, 258, 1001)-net in base 2, because

1 times m-reduction [i] would yield (171, 257, 1001)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 231681 648093 378159 578986 663681 090403 716218 339970 498948 335270 637955 702530 045368 > 2²⁵⁷ [i]