MinT - Best Known (239−210, 239, s)-Nets in Base 3

Best Known (239−210, 239, s)-Nets in Base 3

(239−210, 239, 37)-Net over F₃ — Constructive and digital

Digital (29, 239, 37)-net over F₃, using

t-expansion [i] based on digital (27, 239, 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]

(239−210, 239, 42)-Net over F₃ — Digital

Digital (29, 239, 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]

(239−210, 239, 70)-Net in Base 3 — Upper bound on s

There is no (29, 239, 71)-net in base 3, because

31 times m-reduction [i] would yield (29, 208, 71)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁰⁸, 71, S₃, 3, 179), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 13941 547048 116914 893053 954031 014001 053012 805462 871370 304713 490591 761486 662427 548981 009264 083421 524488 / 5 > 3²⁰⁸ [i]