MinT - Best Known (168−57, 168, s)-Nets in Base 2

Best Known (168−57, 168, s)-Nets in Base 2

(168−57, 168, 68)-Net over F₂ — Constructive and digital

Digital (111, 168, 68)-net over F₂, using

12 times m-reduction [i] based on digital (111, 180, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 90, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(168−57, 168, 84)-Net in Base 2 — Constructive

(111, 168, 84)-net in base 2, using

trace code for nets [i] based on (27, 84, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(168−57, 168, 116)-Net over F₂ — Digital

Digital (111, 168, 116)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−57, 168, 664)-Net in Base 2 — Upper bound on s

There is no (111, 168, 665)-net in base 2, because

1 times m-reduction [i] would yield (111, 167, 665)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 187 234483 064921 301951 423233 262195 499455 349705 389648 > 2¹⁶⁷ [i]