MinT - Best Known (227−197, 227, s)-Nets in Base 3

Best Known (227−197, 227, s)-Nets in Base 3

(227−197, 227, 37)-Net over F₃ — Constructive and digital

Digital (30, 227, 37)-net over F₃, using

t-expansion [i] based on digital (27, 227, 37)-net over F₃, using
- net from sequence [i] based on digital (27, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(227−197, 227, 42)-Net over F₃ — Digital

Digital (30, 227, 42)-net over F₃, using

t-expansion [i] based on digital (29, 227, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(227−197, 227, 72)-Net in Base 3 — Upper bound on s

There is no (30, 227, 73)-net in base 3, because

13 times m-reduction [i] would yield (30, 214, 73)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²¹⁴, 73, S₃, 3, 184), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 354 448149 457943 429626 642095 540246 086021 665939 487358 859705 747695 618620 066719 725201 849556 903898 898410 892351 / 185 > 3²¹⁴ [i]