MinT - Best Known (31, 31+135, s)-Nets in Base 3

Best Known (31, 31+135, s)-Nets in Base 3

(31, 31+135, 37)-Net over F₃ — Constructive and digital

Digital (31, 166, 37)-net over F₃, using

t-expansion [i] based on digital (27, 166, 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]

(31, 31+135, 42)-Net over F₃ — Digital

Digital (31, 166, 42)-net over F₃, using

t-expansion [i] based on digital (29, 166, 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]

(31, 31+135, 79)-Net in Base 3 — Upper bound on s

There is no (31, 166, 80)-net in base 3, because

11 times m-reduction [i] would yield (31, 155, 80)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁵⁵, 80, S₃, 2, 124), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 2427 494450 315468 069358 961833 665581 682507 450011 656972 431201 424883 184770 261689 / 25 > 3¹⁵⁵ [i]