MinT - Best Known (245−112, 245, s)-Nets in Base 2

Best Known (245−112, 245, s)-Nets in Base 2

(245−112, 245, 63)-Net over F₂ — Constructive and digital

Digital (133, 245, 63)-net over F₂, using

4 times m-reduction [i] based on digital (133, 249, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 79, 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, 170, 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]

(245−112, 245, 81)-Net over F₂ — Digital

Digital (133, 245, 81)-net over F₂, using

t-expansion [i] based on digital (126, 245, 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]

(245−112, 245, 298)-Net in Base 2 — Upper bound on s

There is no (133, 245, 299)-net in base 2, because

2 times m-reduction [i] would yield (133, 243, 299)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴³, 299, S₂, 110), but
  - 1 times code embedding in larger space [i] would yield OA(2²⁴⁴, 300, S₂, 110), but
    - the linear programming bound shows that MÂ â‰¥ 727209 932038 995964 963694 786156 506719 769259 430085 807852 102717 045283 079365 167115 254635 995296 956416 / 21378 491075 472167 578125 > 2²⁴⁴ [i]