MinT - Best Known (133, 133+111, s)-Nets in Base 2

Best Known (133, 133+111, s)-Nets in Base 2

(133, 133+111, 66)-Net over F₂ — Constructive and digital

Digital (133, 244, 66)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 94, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (39, 150, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(133, 133+111, 81)-Net over F₂ — Digital

Digital (133, 244, 81)-net over F₂, using

t-expansion [i] based on digital (126, 244, 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]

(133, 133+111, 298)-Net in Base 2 — Upper bound on s

There is no (133, 244, 299)-net in base 2, because

1 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]