MinT - Best Known (78−56, 78, s)-Nets in Base 3

Best Known (78−56, 78, s)-Nets in Base 3

(78−56, 78, 32)-Net over F₃ — Constructive and digital

Digital (22, 78, 32)-net over F₃, using

t-expansion [i] based on digital (21, 78, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(78−56, 78, 73)-Net over F₃ — Upper bound on s (digital)

There is no digital (22, 78, 74)-net over F₃, because

11 times m-reduction [i] would yield digital (22, 67, 74)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁶⁷, 74, F₃, 45) (dual of [74, 7, 46]-code), but
  - residual code [i] would yield linear OA(3²², 28, F₃, 15) (dual of [28, 6, 16]-code), but
    - â€œHHMâ€ bound on codes from Brouwerâ€™s database [i]

(78−56, 78, 75)-Net in Base 3 — Upper bound on s

There is no (22, 78, 76)-net in base 3, because

10 times m-reduction [i] would yield (22, 68, 76)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁶⁸, 76, S₃, 46), but
  - the linear programming bound shows that MÂ â‰¥ 1 013777 979522 262848 532806 654364 768845 / 3619 > 3⁶⁸ [i]