MinT - Best Known (69, 69+85, s)-Nets in Base 2

Best Known (69, 69+85, s)-Nets in Base 2

(69, 69+85, 48)-Net over F₂ — Constructive and digital

Digital (69, 154, 48)-net over F₂, using

net from sequence [i] based on digital (69, 47)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using
  - an explicitly constructive algebraic function field [i]

(69, 69+85, 49)-Net over F₂ — Digital

Digital (69, 154, 49)-net over F₂, using

t-expansion [i] based on digital (68, 154, 49)-net over F₂, using
- net from sequence [i] based on digital (68, 48)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 68 and N(F) ≥ 49, using
    - an algebraic function field by Auer [i]

(69, 69+85, 147)-Net over F₂ — Upper bound on s (digital)

There is no digital (69, 154, 148)-net over F₂, because

13 times m-reduction [i] would yield digital (69, 141, 148)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁴¹, 148, F₂, 72) (dual of [148, 7, 73]-code), but
  - Griesmer bound [i]

(69, 69+85, 149)-Net in Base 2 — Upper bound on s

There is no (69, 154, 150)-net in base 2, because

15 times m-reduction [i] would yield (69, 139, 150)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹³⁹, 150, S₂, 70), but
  - the linear programming bound shows that MÂ â‰¥ 6082 528252 899227 461853 867158 864840 600756 158464 / 8127 > 2¹³⁹ [i]