MinT - Best Known (57−45, 57, s)-Nets in Base 3

Best Known (57−45, 57, s)-Nets in Base 3

(57−45, 57, 20)-Net over F₃ — Constructive and digital

Digital (12, 57, 20)-net over F₃, using

t-expansion [i] based on digital (11, 57, 20)-net over F₃, using
- net from sequence [i] based on digital (11, 19)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]

(57−45, 57, 22)-Net over F₃ — Digital

Digital (12, 57, 22)-net over F₃, using

net from sequence [i] based on digital (12, 21)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 12 and N(F) ≥ 22, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(57−45, 57, 42)-Net over F₃ — Upper bound on s (digital)

There is no digital (12, 57, 43)-net over F₃, because

18 times m-reduction [i] would yield digital (12, 39, 43)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3³⁹, 43, F₃, 27) (dual of [43, 4, 28]-code), but
  - Griesmer bound [i]

(57−45, 57, 45)-Net in Base 3 — Upper bound on s

There is no (12, 57, 46)-net in base 3, because

18 times m-reduction [i] would yield (12, 39, 46)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3³⁹, 46, S₃, 27), but
  - the linear programming bound shows that MÂ â‰¥ 17725 876239 305002 191858 / 4165 > 3³⁹ [i]