MinT - Best Known (254−89, 254, s)-Nets in Base 2

Best Known (254−89, 254, s)-Nets in Base 2

(254−89, 254, 112)-Net over F₂ — Constructive and digital

Digital (165, 254, 112)-net over F₂, using

t-expansion [i] based on digital (163, 254, 112)-net over F₂, using
- 6 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]

(254−89, 254, 153)-Net over F₂ — Digital

Digital (165, 254, 153)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−89, 254, 865)-Net in Base 2 — Upper bound on s

There is no (165, 254, 866)-net in base 2, because

1 times m-reduction [i] would yield (165, 253, 866)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 15053 021620 539671 357695 999525 964162 705428 007896 904929 745870 607477 595096 592368 > 2²⁵³ [i]