MinT - Best Known (115, 115+59, s)-Nets in Base 2

Best Known (115, 115+59, s)-Nets in Base 2

(115, 115+59, 68)-Net over F₂ — Constructive and digital

Digital (115, 174, 68)-net over F₂, using

14 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+59, 84)-Net in Base 2 — Constructive

(115, 174, 84)-net in base 2, using

2 times m-reduction [i] based on (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+59, 120)-Net over F₂ — Digital

Digital (115, 174, 120)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 115+59, 687)-Net in Base 2 — Upper bound on s

There is no (115, 174, 688)-net in base 2, because

1 times m-reduction [i] would yield (115, 173, 688)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 12238 442848 236406 245689 051442 286581 984548 509908 693246 > 2¹⁷³ [i]