MinT - Best Known (31, 31+213, s)-Nets in Base 3

Best Known (31, 31+213, s)-Nets in Base 3

(31, 31+213, 37)-Net over F₃ — Constructive and digital

Digital (31, 244, 37)-net over F₃, using

t-expansion [i] based on digital (27, 244, 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]

(31, 31+213, 42)-Net over F₃ — Digital

Digital (31, 244, 42)-net over F₃, using

t-expansion [i] based on digital (29, 244, 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]

(31, 31+213, 74)-Net in Base 3 — Upper bound on s

There is no (31, 244, 75)-net in base 3, because

24 times m-reduction [i] would yield (31, 220, 75)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²²⁰, 75, S₃, 3, 189), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 25005 745253 694267 115918 266546 661877 100947 851924 479155 683111 942268 320067 287614 161820 805837 058931 962084 566827 / 19 > 3²²⁰ [i]