MinT - Best Known (61−31, 61, s)-Nets in Base 2

Best Known (61−31, 61, s)-Nets in Base 2

(61−31, 61, 22)-Net over F₂ — Constructive and digital

Digital (30, 61, 22)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 21, 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, 40, 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]

(61−31, 61, 25)-Net over F₂ — Digital

Digital (30, 61, 25)-net over F₂, using

t-expansion [i] based on digital (28, 61, 25)-net over F₂, using
- net from sequence [i] based on digital (28, 24)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 28 and N(F) ≥ 25, using
    - a curve from the manYPoints database [i]

(61−31, 61, 71)-Net over F₂ — Upper bound on s (digital)

There is no digital (30, 61, 72)-net over F₂, because

1 times m-reduction [i] would yield digital (30, 60, 72)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶⁰, 72, F₂, 30) (dual of [72, 12, 31]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁶¹, 73, F₂, 31) (dual of [73, 12, 32]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(61−31, 61, 73)-Net in Base 2 — Upper bound on s

There is no (30, 61, 74)-net in base 2, because

1 times m-reduction [i] would yield (30, 60, 74)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁰, 74, S₂, 30), but
  - the linear programming bound shows that MÂ â‰¥ 117 886223 846050 103296 / 95 > 2⁶⁰ [i]