MinT - Best Known (58−29, 58, s)-Nets in Base 2

Best Known (58−29, 58, s)-Nets in Base 2

(58−29, 58, 22)-Net over F₂ — Constructive and digital

Digital (29, 58, 22)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 20, 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, 38, 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]

(58−29, 58, 25)-Net over F₂ — Digital

Digital (29, 58, 25)-net over F₂, using

t-expansion [i] based on digital (28, 58, 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]

(58−29, 58, 69)-Net over F₂ — Upper bound on s (digital)

There is no digital (29, 58, 70)-net over F₂, because

1 times m-reduction [i] would yield digital (29, 57, 70)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁷, 70, F₂, 28) (dual of [70, 13, 29]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁵⁸, 71, F₂, 29) (dual of [71, 13, 30]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(58−29, 58, 72)-Net in Base 2 — Upper bound on s

There is no (29, 58, 73)-net in base 2, because

1 times m-reduction [i] would yield (29, 57, 73)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁷, 73, S₂, 28), but
  - the linear programming bound shows that MÂ â‰¥ 10440 857145 719606 214656 / 72105 > 2⁵⁷ [i]