MinT - Best Known (65−36, 65, s)-Nets in Base 2

Best Known (65−36, 65, s)-Nets in Base 2

(65−36, 65, 21)-Net over F₂ — Constructive and digital

Digital (29, 65, 21)-net over F₂, using

t-expansion [i] based on digital (21, 65, 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]

(65−36, 65, 25)-Net over F₂ — Digital

Digital (29, 65, 25)-net over F₂, using

t-expansion [i] based on digital (28, 65, 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]

(65−36, 65, 65)-Net over F₂ — Upper bound on s (digital)

There is no digital (29, 65, 66)-net over F₂, because

4 times m-reduction [i] would yield digital (29, 61, 66)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶¹, 66, F₂, 32) (dual of [66, 5, 33]-code), but
  - Griesmer bound [i]

(65−36, 65, 67)-Net in Base 2 — Upper bound on s

There is no (29, 65, 68)-net in base 2, because

2 times m-reduction [i] would yield (29, 63, 68)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶³, 68, S₂, 34), but
  - adding a parity check bit [i] would yield OA(2⁶⁴, 69, S₂, 35), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 73 786976 294838 206464 / 3 > 2⁶⁴ [i]