MinT - Best Known (31, 64, s)-Nets in Base 2

Best Known (31, 64, s)-Nets in Base 2

(31, 64, 22)-Net over F₂ — Constructive and digital

Digital (31, 64, 22)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 22, 10)-net over F₂, using
  - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (9, 42, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]

(31, 64, 27)-Net over F₂ — Digital

Digital (31, 64, 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, 64, 70)-Net over F₂ — Upper bound on s (digital)

There is no digital (31, 64, 71)-net over F₂, because

1 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, 64, 72)-Net in Base 2 — Upper bound on s

There is no (31, 64, 73)-net in base 2, because

1 times m-reduction [i] would yield (31, 63, 73)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶³, 73, S₂, 32), but
  - the linear programming bound shows that MÂ â‰¥ 12101 064112 353465 860096 / 1309 > 2⁶³ [i]