MinT - Best Known (60−28, 60, s)-Nets in Base 2

Best Known (60−28, 60, s)-Nets in Base 2

(60−28, 60, 24)-Net over F₂ — Constructive and digital

Digital (32, 60, 24)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 23, 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]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (9, 37, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂ (see above)

(60−28, 60, 27)-Net over F₂ — Digital

Digital (32, 60, 27)-net over F₂, using

t-expansion [i] based on digital (31, 60, 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]

(60−28, 60, 90)-Net over F₂ — Upper bound on s (digital)

There is no digital (32, 60, 91)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁶⁰, 91, F₂, 28) (dual of [91, 31, 29]-code), but
- adding a parity check bit [i] would yield linear OA(2⁶¹, 92, F₂, 29) (dual of [92, 31, 30]-code), but
  - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(60−28, 60, 91)-Net in Base 2 — Upper bound on s

There is no (32, 60, 92)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁶⁰, 92, S₂, 28), but
- the linear programming bound shows that MÂ â‰¥ 2328 864103 095825 703387 529216 / 1748 342475 > 2⁶⁰ [i]