MinT - Best Known (257−79, 257, s)-Nets in Base 2

Best Known (257−79, 257, s)-Nets in Base 2

(257−79, 257, 132)-Net over F₂ — Constructive and digital

Digital (178, 257, 132)-net over F₂, using

1 times m-reduction [i] based on digital (178, 258, 132)-net over F₂, using
- trace code for nets [i] based on digital (49, 129, 66)-net over F₄, using
  - net from sequence [i] based on digital (49, 65)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
      - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(257−79, 257, 209)-Net over F₂ — Digital

Digital (178, 257, 209)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−79, 257, 1399)-Net in Base 2 — Upper bound on s

There is no (178, 257, 1400)-net in base 2, because

1 times m-reduction [i] would yield (178, 256, 1400)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 116123 721930 855960 238483 741091 329701 336491 489388 010081 311140 593415 845379 440827 > 2²⁵⁶ [i]