MinT - Best Known (164−137, 164, s)-Nets in Base 2

Best Known (164−137, 164, s)-Nets in Base 2

(164−137, 164, 21)-Net over F₂ — Constructive and digital

Digital (27, 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]

(164−137, 164, 24)-Net over F₂ — Digital

Digital (27, 164, 24)-net over F₂, using

t-expansion [i] based on digital (25, 164, 24)-net over F₂, using
- net from sequence [i] based on digital (25, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
    - Drinfeld modules of rank 1 [i]

(164−137, 164, 36)-Net in Base 2 — Upper bound on s

There is no (27, 164, 37)-net in base 2, because

25 times m-reduction [i] would yield (27, 139, 37)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹³⁹, 37, S₂, 4, 112), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 101 747149 968295 968083 256714 618958 149371 035648 / 113 > 2¹³⁹ [i]