MinT - Best Known (17, 36, s)-Nets in Base 2

Best Known (17, 36, s)-Nets in Base 2

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

Digital (17, 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]

(17, 36, 42)-Net over F₂ — Upper bound on s (digital)

There is no digital (17, 36, 43)-net over F₂, because

1 times m-reduction [i] would yield digital (17, 35, 43)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2³⁵, 43, F₂, 18) (dual of [43, 8, 19]-code), but
  - adding a parity check bit [i] would yield linear OA(2³⁶, 44, F₂, 19) (dual of [44, 8, 20]-code), but
    - â€œDMâ€ bound on codes from Brouwerâ€™s database [i]

(17, 36, 46)-Net in Base 2 — Upper bound on s

There is no (17, 36, 47)-net in base 2, because

1 times m-reduction [i] would yield (17, 35, 47)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2³⁵, 47, S₂, 18), but
  - the linear programming bound shows that MÂ â‰¥ 136 339441 844224 / 3705 > 2³⁵ [i]