MinT - Best Known (168−136, 168, s)-Nets in Base 3

Best Known (168−136, 168, s)-Nets in Base 3

(168−136, 168, 38)-Net over F₃ — Constructive and digital

Digital (32, 168, 38)-net over F₃, using

net from sequence [i] based on digital (32, 37)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
  - an explicitly constructive algebraic function field [i]

(168−136, 168, 42)-Net over F₃ — Digital

Digital (32, 168, 42)-net over F₃, using

t-expansion [i] based on digital (29, 168, 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]

(168−136, 168, 82)-Net in Base 3 — Upper bound on s

There is no (32, 168, 83)-net in base 3, because

8 times m-reduction [i] would yield (32, 160, 83)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶⁰, 83, S₃, 2, 128), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 1 114219 952694 799843 835763 481652 501992 270919 555350 550345 921454 021381 809550 115251 / 43 > 3¹⁶⁰ [i]