MinT - Best Known (123, 123+122, s)-Nets in Base 2

Best Known (123, 123+122, s)-Nets in Base 2

(123, 123+122, 57)-Net over F₂ — Constructive and digital

Digital (123, 245, 57)-net over F₂, using

t-expansion [i] based on digital (110, 245, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(123, 123+122, 80)-Net over F₂ — Digital

Digital (123, 245, 80)-net over F₂, using

t-expansion [i] based on digital (121, 245, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(123, 123+122, 260)-Net over F₂ — Upper bound on s (digital)

There is no digital (123, 245, 261)-net over F₂, because

2 times m-reduction [i] would yield digital (123, 243, 261)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁴³, 261, F₂, 120) (dual of [261, 18, 121]-code), but
  - residual code [i] would yield OA(2¹²³, 140, S₂, 60), but
    - the linear programming bound shows that MÂ â‰¥ 3 888746 889172 484760 459445 013730 247120 519168 / 329189 > 2¹²³ [i]

(123, 123+122, 262)-Net in Base 2 — Upper bound on s

There is no (123, 245, 263)-net in base 2, because

12 times m-reduction [i] would yield (123, 233, 263)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³³, 263, S₂, 110), but
  - the linear programming bound shows that MÂ â‰¥ 208 361079 889821 744668 890387 660731 037374 858559 867656 192612 204735 454426 713127 124992 / 13955 210325 > 2²³³ [i]