MinT - Best Known (254−122, 254, s)-Nets in Base 2

Best Known (254−122, 254, s)-Nets in Base 2

(254−122, 254, 59)-Net over F₂ — Constructive and digital

Digital (132, 254, 59)-net over F₂, using

4 times m-reduction [i] based on digital (132, 258, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 78, 17)-net over F₂, using
    - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 180, 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]

(254−122, 254, 81)-Net over F₂ — Digital

Digital (132, 254, 81)-net over F₂, using

t-expansion [i] based on digital (126, 254, 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]

(254−122, 254, 296)-Net in Base 2 — Upper bound on s

There is no (132, 254, 297)-net in base 2, because

14 times m-reduction [i] would yield (132, 240, 297)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴⁰, 297, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 28020 398795 768772 015654 276848 365160 205592 563190 101250 876110 646415 347830 039205 258520 634937 311232 / 15527 302685 925212 739375 > 2²⁴⁰ [i]