MinT - Best Known (138−112, 138, s)-Nets in Base 3

Best Known (138−112, 138, s)-Nets in Base 3

(138−112, 138, 36)-Net over F₃ — Constructive and digital

Digital (26, 138, 36)-net over F₃, using

net from sequence [i] based on digital (26, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
  - an explicitly constructive algebraic function field [i]

(138−112, 138, 37)-Net over F₃ — Digital

Digital (26, 138, 37)-net over F₃, using

net from sequence [i] based on digital (26, 36)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 25, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(138−112, 138, 68)-Net in Base 3 — Upper bound on s

There is no (26, 138, 69)-net in base 3, because

6 times m-reduction [i] would yield (26, 132, 69)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹³², 69, S₃, 2, 106), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 111735 579243 228552 716503 312107 606439 394632 411259 998146 190911 899397 / 107 > 3¹³² [i]