MinT - Best Known (257−87, 257, s)-Nets in Base 2

Best Known (257−87, 257, s)-Nets in Base 2

(257−87, 257, 112)-Net over F₂ — Constructive and digital

Digital (170, 257, 112)-net over F₂, using

t-expansion [i] based on digital (163, 257, 112)-net over F₂, using
- 3 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]

(257−87, 257, 167)-Net over F₂ — Digital

Digital (170, 257, 167)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−87, 257, 984)-Net in Base 2 — Upper bound on s

There is no (170, 257, 985)-net in base 2, because

1 times m-reduction [i] would yield (170, 256, 985)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 120585 851482 910857 870739 236858 701880 421952 256722 491116 284831 254416 057177 768080 > 2²⁵⁶ [i]