MinT - Best Known (245−77, 245, s)-Nets in Base 2

Best Known (245−77, 245, s)-Nets in Base 2

(245−77, 245, 112)-Net over F₂ — Constructive and digital

Digital (168, 245, 112)-net over F₂, using

t-expansion [i] based on digital (163, 245, 112)-net over F₂, using
- 15 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]

(245−77, 245, 190)-Net over F₂ — Digital

Digital (168, 245, 190)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−77, 245, 1232)-Net in Base 2 — Upper bound on s

There is no (168, 245, 1233)-net in base 2, because

1 times m-reduction [i] would yield (168, 244, 1233)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 29 059929 953162 936655 244795 572015 391353 615031 838661 094948 414121 162490 239664 > 2²⁴⁴ [i]