MinT - Best Known (230−103, 230, s)-Nets in Base 2

Best Known (230−103, 230, s)-Nets in Base 2

(230−103, 230, 63)-Net over F₂ — Constructive and digital

Digital (127, 230, 63)-net over F₂, using

1 times m-reduction [i] based on digital (127, 231, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 73, 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, 158, 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]

(230−103, 230, 81)-Net over F₂ — Digital

Digital (127, 230, 81)-net over F₂, using

t-expansion [i] based on digital (126, 230, 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]

(230−103, 230, 292)-Net in Base 2 — Upper bound on s

There is no (127, 230, 293)-net in base 2, because

1 times m-reduction [i] would yield (127, 229, 293)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁹, 293, S₂, 102), but
  - the linear programming bound shows that MÂ â‰¥ 8205 062837 078930 532068 639436 021052 815868 020242 277045 458586 682301 189324 495527 824585 516900 155392 / 7 988981 954712 652237 026475 > 2²²⁹ [i]