MinT - Best Known (72−41, 72, s)-Nets in Base 2

Best Known (72−41, 72, s)-Nets in Base 2

(72−41, 72, 21)-Net over F₂ — Constructive and digital

Digital (31, 72, 21)-net over F₂, using

t-expansion [i] based on digital (21, 72, 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]

(72−41, 72, 27)-Net over F₂ — Digital

Digital (31, 72, 27)-net over F₂, using

net from sequence [i] based on digital (31, 26)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 31 and N(F) ≥ 27, using
  - Drinfeld modules of rank 1 [i]

(72−41, 72, 70)-Net over F₂ — Upper bound on s (digital)

There is no digital (31, 72, 71)-net over F₂, because

9 times m-reduction [i] would yield digital (31, 63, 71)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶³, 71, F₂, 32) (dual of [71, 8, 33]-code), but
  - residual code [i] would yield linear OA(2³¹, 38, F₂, 16) (dual of [38, 7, 17]-code), but
    - residual code [i] would yield linear OA(2¹⁵, 21, F₂, 8) (dual of [21, 6, 9]-code), but
      - residual code [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
        bound by Fontaine and Peterson [i]

(72−41, 72, 71)-Net in Base 2 — Upper bound on s

There is no (31, 72, 72)-net in base 2, because

1 times m-reduction [i] would yield (31, 71, 72)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2749 939722 480302 234756 > 2⁷¹ [i]