MinT - Best Known (164−135, 164, s)-Nets in Base 2

Best Known (164−135, 164, s)-Nets in Base 2

(164−135, 164, 21)-Net over F₂ — Constructive and digital

Digital (29, 164, 21)-net over F₂, using

t-expansion [i] based on digital (21, 164, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(164−135, 164, 25)-Net over F₂ — Digital

Digital (29, 164, 25)-net over F₂, using

t-expansion [i] based on digital (28, 164, 25)-net over F₂, using
- net from sequence [i] based on digital (28, 24)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 28 and N(F) ≥ 25, using
    - a curve from the manYPoints database [i]

(164−135, 164, 38)-Net in Base 2 — Upper bound on s

There is no (29, 164, 39)-net in base 2, because

17 times m-reduction [i] would yield (29, 147, 39)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁴⁷, 39, S₂, 4, 118), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 23371 180968 060093 052329 432749 735482 858267 475968 / 119 > 2¹⁴⁷ [i]