MinT - Best Known (238−110, 238, s)-Nets in Base 2

Best Known (238−110, 238, s)-Nets in Base 2

(238−110, 238, 62)-Net over F₂ — Constructive and digital

Digital (128, 238, 62)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 74, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 164, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(238−110, 238, 81)-Net over F₂ — Digital

Digital (128, 238, 81)-net over F₂, using

t-expansion [i] based on digital (126, 238, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(238−110, 238, 279)-Net in Base 2 — Upper bound on s

There is no (128, 238, 280)-net in base 2, because

2 times m-reduction [i] would yield (128, 236, 280)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁶, 280, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 41504 278412 335844 454047 491725 650645 432186 567323 902384 277507 042637 995046 295745 131031 035904 / 340479 332843 203125 > 2²³⁶ [i]