MinT - Best Known (12, 12+137, s)-Nets in Base 3

Best Known (12, 12+137, s)-Nets in Base 3

(12, 12+137, 20)-Net over F₃ — Constructive and digital

Digital (12, 149, 20)-net over F₃, using

t-expansion [i] based on digital (11, 149, 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]

(12, 12+137, 22)-Net over F₃ — Digital

Digital (12, 149, 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]

(12, 12+137, 33)-Net in Base 3 — Upper bound on s

There is no (12, 149, 34)-net in base 3, because

52 times m-reduction [i] would yield (12, 97, 34)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁹⁷, 34, S₃, 3, 85), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 858962 534553 352218 394101 882942 702121 170179 203335 / 43 > 3⁹⁷ [i]