MinT - Best Known (177−61, 177, s)-Nets in Base 2

Best Known (177−61, 177, s)-Nets in Base 2

(177−61, 177, 68)-Net over F₂ — Constructive and digital

Digital (116, 177, 68)-net over F₂, using

13 times m-reduction [i] based on digital (116, 190, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 95, 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]

(177−61, 177, 84)-Net in Base 2 — Constructive

(116, 177, 84)-net in base 2, using

1 times m-reduction [i] based on (116, 178, 84)-net in base 2, using
- trace code for nets [i] based on (27, 89, 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]

(177−61, 177, 117)-Net over F₂ — Digital

Digital (116, 177, 117)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(177−61, 177, 659)-Net in Base 2 — Upper bound on s

There is no (116, 177, 660)-net in base 2, because

1 times m-reduction [i] would yield (116, 176, 660)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 97665 841609 849788 692277 866086 787548 396172 004810 660064 > 2¹⁷⁶ [i]