MinT - Best Known (115, 115+61, s)-Nets in Base 2

Best Known (115, 115+61, s)-Nets in Base 2

(115, 115+61, 68)-Net over F₂ — Constructive and digital

Digital (115, 176, 68)-net over F₂, using

12 times m-reduction [i] based on digital (115, 188, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 94, 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]

(115, 115+61, 84)-Net in Base 2 — Constructive

(115, 176, 84)-net in base 2, using

trace code for nets [i] based on (27, 88, 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]

(115, 115+61, 115)-Net over F₂ — Digital

Digital (115, 176, 115)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 115+61, 643)-Net in Base 2 — Upper bound on s

There is no (115, 176, 644)-net in base 2, because

1 times m-reduction [i] would yield (115, 175, 644)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 48906 339477 353909 471223 005846 927058 279113 350894 644000 > 2¹⁷⁵ [i]