MinT - Best Known (78−66, 78, s)-Nets in Base 3

Best Known (78−66, 78, s)-Nets in Base 3

(78−66, 78, 20)-Net over F₃ — Constructive and digital

Digital (12, 78, 20)-net over F₃, using

t-expansion [i] based on digital (11, 78, 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]

(78−66, 78, 22)-Net over F₃ — Digital

Digital (12, 78, 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]

(78−66, 78, 35)-Net in Base 3 — Upper bound on s

There is no (12, 78, 36)-net in base 3, because

11 times m-reduction [i] would yield (12, 67, 36)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁶⁷, 36, S₃, 2, 55), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 5284 439399 430176 713888 429748 403459 / 56 > 3⁶⁷ [i]