MinT - Best Known (62−29, 62, s)-Nets in Base 2

Best Known (62−29, 62, s)-Nets in Base 2

(62−29, 62, 25)-Net over F₂ — Constructive and digital

Digital (33, 62, 25)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 22, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (11, 40, 14)-net over F₂, using
  - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]

(62−29, 62, 28)-Net over F₂ — Digital

Digital (33, 62, 28)-net over F₂, using

net from sequence [i] based on digital (33, 27)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 28, using
  - Drinfeld modules of rank 1 [i]

(62−29, 62, 99)-Net over F₂ — Upper bound on s (digital)

There is no digital (33, 62, 100)-net over F₂, because

1 times m-reduction [i] would yield digital (33, 61, 100)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶¹, 100, F₂, 28) (dual of [100, 39, 29]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁶², 101, F₂, 29) (dual of [101, 39, 30]-code), but
    - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(62−29, 62, 100)-Net in Base 2 — Upper bound on s

There is no (33, 62, 101)-net in base 2, because

1 times m-reduction [i] would yield (33, 61, 101)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶¹, 101, S₂, 28), but
  - the linear programming bound shows that MÂ â‰¥ 6546 972925 691403 581923 536083 615744 / 2497 320966 828051 > 2⁶¹ [i]