MinT - Best Known (169−57, 169, s)-Nets in Base 2

Best Known (169−57, 169, s)-Nets in Base 2

(169−57, 169, 68)-Net over F₂ — Constructive and digital

Digital (112, 169, 68)-net over F₂, using

13 times m-reduction [i] based on digital (112, 182, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 91, 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]

(169−57, 169, 84)-Net in Base 2 — Constructive

(112, 169, 84)-net in base 2, using

1 times m-reduction [i] based on (112, 170, 84)-net in base 2, using
- trace code for nets [i] based on (27, 85, 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]

(169−57, 169, 118)-Net over F₂ — Digital

Digital (112, 169, 118)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−57, 169, 682)-Net in Base 2 — Upper bound on s

There is no (112, 169, 683)-net in base 2, because

1 times m-reduction [i] would yield (112, 168, 683)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 379 508120 419219 337822 520547 815960 724307 531241 876072 > 2¹⁶⁸ [i]