MinT - Best Known (38, 81, s)-Nets in Base 2

Best Known (38, 81, s)-Nets in Base 2

(38, 81, 24)-Net over F₂ — Constructive and digital

Digital (38, 81, 24)-net over F₂, using

t-expansion [i] based on digital (33, 81, 24)-net over F₂, using
- net from sequence [i] based on digital (33, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(38, 81, 30)-Net over F₂ — Digital

Digital (38, 81, 30)-net over F₂, using

t-expansion [i] based on digital (36, 81, 30)-net over F₂, using
- net from sequence [i] based on digital (36, 29)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
    - Drinfeld modules of rank 1 [i]

(38, 81, 84)-Net over F₂ — Upper bound on s (digital)

There is no digital (38, 81, 85)-net over F₂, because

3 times m-reduction [i] would yield digital (38, 78, 85)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷⁸, 85, F₂, 40) (dual of [85, 7, 41]-code), but
  - â€œHelâ€ bound on codes from Brouwerâ€™s database [i]

(38, 81, 87)-Net in Base 2 — Upper bound on s

There is no (38, 81, 88)-net in base 2, because

3 times m-reduction [i] would yield (38, 78, 88)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷⁸, 88, S₂, 40), but
  - the linear programming bound shows that MÂ â‰¥ 2795 036494 949022 651920 678912 / 7371 > 2⁷⁸ [i]