MinT - Best Known (170−51, 170, s)-Nets in Base 2

Best Known (170−51, 170, s)-Nets in Base 2

(170−51, 170, 112)-Net over F₂ — Constructive and digital

Digital (119, 170, 112)-net over F₂, using

2 times m-reduction [i] based on digital (119, 172, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 86, 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]

(170−51, 170, 156)-Net over F₂ — Digital

Digital (119, 170, 156)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−51, 170, 1066)-Net in Base 2 — Upper bound on s

There is no (119, 170, 1067)-net in base 2, because

1 times m-reduction [i] would yield (119, 169, 1067)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 755 280669 387578 509095 809319 881384 636445 866050 592904 > 2¹⁶⁹ [i]