MinT - Best Known (4, 4+11, s)-Nets in Base 3

Best Known (4, 4+11, s)-Nets in Base 3

(4, 4+11, 12)-Net over F₃ — Constructive and digital

Digital (4, 15, 12)-net over F₃, using

net from sequence [i] based on digital (4, 11)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
  - an explicitly constructive algebraic function field [i]

(4, 4+11, 18)-Net over F₃ — Upper bound on s (digital)

There is no digital (4, 15, 19)-net over F₃, because

2 times m-reduction [i] would yield digital (4, 13, 19)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹³, 19, F₃, 9) (dual of [19, 6, 10]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹², 15, F₃, 9) (dual of [15, 3, 10]-code), but
      - Griesmer bound [i]
    2. OA(3⁶, 19, S₃, 4), but
      - the linear programming bound shows that MÂ â‰¥ 51975 / 67 > 3⁶ [i]

(4, 4+11, 21)-Net in Base 3 — Upper bound on s

There is no (4, 15, 22)-net in base 3, because

extracting embedded OOA [i] would yield OOA(3¹⁵, 22, S₃, 2, 11), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 20 073575 120656 593609 / 1 302201 183296 > 3¹⁵ [i]