MinT - Best Known (253−89, 253, s)-Nets in Base 2

Best Known (253−89, 253, s)-Nets in Base 2

(253−89, 253, 112)-Net over F₂ — Constructive and digital

Digital (164, 253, 112)-net over F₂, using

t-expansion [i] based on digital (163, 253, 112)-net over F₂, using
- 7 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]

(253−89, 253, 151)-Net over F₂ — Digital

Digital (164, 253, 151)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−89, 253, 850)-Net in Base 2 — Upper bound on s

There is no (164, 253, 851)-net in base 2, because

1 times m-reduction [i] would yield (164, 252, 851)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7345 978706 512210 816992 086433 530150 942901 230644 721032 905307 358367 317118 303768 > 2²⁵² [i]