MinT - Best Known (36−18, 36, s)-Nets in Base 2

Best Known (36−18, 36, s)-Nets in Base 2

(36−18, 36, 17)-Net over F₂ — Constructive and digital

Digital (18, 36, 17)-net over F₂, using

t-expansion [i] based on digital (15, 36, 17)-net over F₂, using
- net from sequence [i] based on digital (15, 16)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(36−18, 36, 18)-Net over F₂ — Digital

Digital (18, 36, 18)-net over F₂, using

net from sequence [i] based on digital (18, 17)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 18 and N(F) ≥ 18, using
  - a curve from the manYPoints database [i]

(36−18, 36, 46)-Net over F₂ — Upper bound on s (digital)

There is no digital (18, 36, 47)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2³⁶, 47, F₂, 18) (dual of [47, 11, 19]-code), but
- adding a parity check bit [i] would yield linear OA(2³⁷, 48, F₂, 19) (dual of [48, 11, 20]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(36−18, 36, 52)-Net in Base 2 — Upper bound on s

There is no (18, 36, 53)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2³⁶, 53, S₂, 18), but
- the linear programming bound shows that MÂ â‰¥ 20 856361 189376 / 247 > 2³⁶ [i]