MinT - Best Known (67−37, 67, s)-Nets in Base 2

Best Known (67−37, 67, s)-Nets in Base 2

(67−37, 67, 21)-Net over F₂ — Constructive and digital

Digital (30, 67, 21)-net over F₂, using

t-expansion [i] based on digital (21, 67, 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]

(67−37, 67, 25)-Net over F₂ — Digital

Digital (30, 67, 25)-net over F₂, using

t-expansion [i] based on digital (28, 67, 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]

(67−37, 67, 67)-Net over F₂ — Upper bound on s (digital)

There is no digital (30, 67, 68)-net over F₂, because

5 times m-reduction [i] would yield digital (30, 62, 68)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶², 68, F₂, 32) (dual of [68, 6, 33]-code), but
  - Griesmer bound [i]

(67−37, 67, 69)-Net in Base 2 — Upper bound on s

There is no (30, 67, 70)-net in base 2, because

1 times m-reduction [i] would yield (30, 66, 70)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁶, 70, S₂, 36), but
  - adding a parity check bit [i] would yield OA(2⁶⁷, 71, S₂, 37), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 2951 479051 793528 258560 / 19 > 2⁶⁷ [i]