MinT - Best Known (240−109, 240, s)-Nets in Base 2

Best Known (240−109, 240, s)-Nets in Base 2

(240−109, 240, 63)-Net over F₂ — Constructive and digital

Digital (131, 240, 63)-net over F₂, using

3 times m-reduction [i] based on digital (131, 243, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 77, 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]
  2. digital (54, 166, 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]

(240−109, 240, 81)-Net over F₂ — Digital

Digital (131, 240, 81)-net over F₂, using

t-expansion [i] based on digital (126, 240, 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]

(240−109, 240, 291)-Net in Base 2 — Upper bound on s

There is no (131, 240, 292)-net in base 2, because

1 times m-reduction [i] would yield (131, 239, 292)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁹, 292, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 129 753212 264703 737764 759763 826530 014325 194921 459040 258227 297024 189374 782768 881401 211050 262528 / 138 951361 618522 578125 > 2²³⁹ [i]