MinT - Best Known (164−49, 164, s)-Nets in Base 2

Best Known (164−49, 164, s)-Nets in Base 2

(164−49, 164, 112)-Net over F₂ — Constructive and digital

Digital (115, 164, 112)-net over F₂, using

trace code for nets [i] based on digital (33, 82, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(164−49, 164, 153)-Net over F₂ — Digital

Digital (115, 164, 153)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−49, 164, 1051)-Net in Base 2 — Upper bound on s

There is no (115, 164, 1052)-net in base 2, because

1 times m-reduction [i] would yield (115, 163, 1052)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 11 929231 079955 336843 414237 495212 439305 263713 240493 > 2¹⁶³ [i]