MinT - Best Known (247−75, 247, s)-Nets in Base 2

Best Known (247−75, 247, s)-Nets in Base 2

(247−75, 247, 112)-Net over F₂ — Constructive and digital

Digital (172, 247, 112)-net over F₂, using

t-expansion [i] based on digital (163, 247, 112)-net over F₂, using
- 13 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]

(247−75, 247, 207)-Net over F₂ — Digital

Digital (172, 247, 207)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−75, 247, 1416)-Net in Base 2 — Upper bound on s

There is no (172, 247, 1417)-net in base 2, because

1 times m-reduction [i] would yield (172, 246, 1417)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 115 815901 652990 793794 269122 990175 210246 563819 137876 624183 872686 070275 357456 > 2²⁴⁶ [i]