MinT - Best Known (77−56, 77, s)-Nets in Base 3

Best Known (77−56, 77, s)-Nets in Base 3

(77−56, 77, 32)-Net over F₃ — Constructive and digital

Digital (21, 77, 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]

(77−56, 77, 70)-Net over F₃ — Upper bound on s (digital)

There is no digital (21, 77, 71)-net over F₃, because

11 times m-reduction [i] would yield digital (21, 66, 71)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁶⁶, 71, F₃, 45) (dual of [71, 5, 46]-code), but
  - â€œHW1â€ bound on codes from Brouwerâ€™s database [i]

(77−56, 77, 73)-Net in Base 3 — Upper bound on s

There is no (21, 77, 74)-net in base 3, because

14 times m-reduction [i] would yield (21, 63, 74)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁶³, 74, S₃, 42), but
  - the linear programming bound shows that MÂ â‰¥ 12 034337 282839 174538 591498 251772 066709 / 10 069525 > 3⁶³ [i]