MinT - Best Known (172−49, 172, s)-Nets in Base 2

Best Known (172−49, 172, s)-Nets in Base 2

(172−49, 172, 112)-Net over F₂ — Constructive and digital

Digital (123, 172, 112)-net over F₂, using

8 times m-reduction [i] based on digital (123, 180, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 90, 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]

(172−49, 172, 178)-Net over F₂ — Digital

Digital (123, 172, 178)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−49, 172, 1333)-Net in Base 2 — Upper bound on s

There is no (123, 172, 1334)-net in base 2, because

1 times m-reduction [i] would yield (123, 171, 1334)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3031 300651 921760 612516 579181 917726 555196 350314 666436 > 2¹⁷¹ [i]