MinT - Best Known (31, 31+37, s)-Nets in Base 2

Best Known (31, 31+37, s)-Nets in Base 2

(31, 31+37, 21)-Net over F₂ — Constructive and digital

Digital (31, 68, 21)-net over F₂, using

t-expansion [i] based on digital (21, 68, 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]

(31, 31+37, 27)-Net over F₂ — Digital

Digital (31, 68, 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]

(31, 31+37, 70)-Net over F₂ — Upper bound on s (digital)

There is no digital (31, 68, 71)-net over F₂, because

5 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]

(31, 31+37, 71)-Net in Base 2 — Upper bound on s

There is no (31, 68, 72)-net in base 2, because

1 times m-reduction [i] would yield (31, 67, 72)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁷, 72, S₂, 36), but
  - adding a parity check bit [i] would yield OA(2⁶⁸, 73, S₂, 37), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 7083 549724 304467 820544 / 19 > 2⁶⁸ [i]