MinT - Best Known (257−77, 257, s)-Nets in Base 2

Best Known (257−77, 257, s)-Nets in Base 2

(257−77, 257, 132)-Net over F₂ — Constructive and digital

Digital (180, 257, 132)-net over F₂, using

t-expansion [i] based on digital (179, 257, 132)-net over F₂, using
- 3 times m-reduction [i] based on digital (179, 260, 132)-net over F₂, using
  - trace code for nets [i] based on digital (49, 130, 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]

(257−77, 257, 222)-Net over F₂ — Digital

Digital (180, 257, 222)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−77, 257, 1546)-Net in Base 2 — Upper bound on s

There is no (180, 257, 1547)-net in base 2, because

1 times m-reduction [i] would yield (180, 256, 1547)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 116112 211192 953433 156342 709092 317416 509751 702459 011560 016656 351515 136682 395680 > 2²⁵⁶ [i]