MinT - Best Known (172−53, 172, s)-Nets in Base 2

Best Known (172−53, 172, s)-Nets in Base 2

(172−53, 172, 112)-Net over F₂ — Constructive and digital

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]

(172−53, 172, 148)-Net over F₂ — Digital

Digital (119, 172, 148)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−53, 172, 969)-Net in Base 2 — Upper bound on s

There is no (119, 172, 970)-net in base 2, because

1 times m-reduction [i] would yield (119, 171, 970)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3037 974909 975012 750718 637992 899004 269096 555881 668480 > 2¹⁷¹ [i]