MinT - Best Known (42−21, 42, s)-Nets in Base 2

Best Known (42−21, 42, s)-Nets in Base 2

(42−21, 42, 21)-Net over F₂ — Constructive and digital

Digital (21, 42, 21)-net over F₂, using

net from sequence [i] based on digital (21, 20)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
  - an explicitly constructive algebraic function field [i]

(42−21, 42, 53)-Net over F₂ — Upper bound on s (digital)

There is no digital (21, 42, 54)-net over F₂, because

1 times m-reduction [i] would yield digital (21, 41, 54)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁴¹, 54, F₂, 20) (dual of [54, 13, 21]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁴², 55, F₂, 21) (dual of [55, 13, 22]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(42−21, 42, 61)-Net in Base 2 — Upper bound on s

There is no (21, 42, 62)-net in base 2, because

1 times m-reduction [i] would yield (21, 41, 62)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁴¹, 62, S₂, 20), but
  - the linear programming bound shows that MÂ â‰¥ 249180 121179 619328 / 104975 > 2⁴¹ [i]