MinT - Best Known (37, 68, s)-Nets in Base 2

Best Known (37, 68, s)-Nets in Base 2

(37, 68, 28)-Net over F₂ — Constructive and digital

Digital (37, 68, 28)-net over F₂, using

trace code for nets [i] based on digital (3, 34, 14)-net over F₄, using
- net from sequence [i] based on digital (3, 13)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
    - an explicitly constructive algebraic function field [i]

(37, 68, 30)-Net over F₂ — Digital

Digital (37, 68, 30)-net over F₂, using

t-expansion [i] based on digital (36, 68, 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]

(37, 68, 115)-Net over F₂ — Upper bound on s (digital)

There is no digital (37, 68, 116)-net over F₂, because

1 times m-reduction [i] would yield digital (37, 67, 116)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶⁷, 116, F₂, 30) (dual of [116, 49, 31]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁶⁸, 117, F₂, 31) (dual of [117, 49, 32]-code), but
    - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(37, 68, 116)-Net in Base 2 — Upper bound on s

There is no (37, 68, 117)-net in base 2, because

1 times m-reduction [i] would yield (37, 67, 117)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁷, 117, S₂, 30), but
  - the linear programming bound shows that MÂ â‰¥ 378539 140513 856120 853549 293619 966386 372608 / 2507 058247 231745 266941 > 2⁶⁷ [i]