MinT - Best Known (157−128, 157, s)-Nets in Base 3

Best Known (157−128, 157, s)-Nets in Base 3

(157−128, 157, 37)-Net over F₃ — Constructive and digital

Digital (29, 157, 37)-net over F₃, using

t-expansion [i] based on digital (27, 157, 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]

(157−128, 157, 42)-Net over F₃ — Digital

Digital (29, 157, 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]

(157−128, 157, 75)-Net in Base 3 — Upper bound on s

There is no (29, 157, 76)-net in base 3, because

11 times m-reduction [i] would yield (29, 146, 76)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁴⁶, 76, S₃, 2, 117), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 616647 475058 544954 874501 304086 161073 644121 833982 871623 025307 342169 580415 / 118 > 3¹⁴⁶ [i]