MinT - Best Known (30, 30+49, s)-Nets in Base 3

Best Known (30, 30+49, s)-Nets in Base 3

(30, 30+49, 37)-Net over F₃ — Constructive and digital

Digital (30, 79, 37)-net over F₃, using

t-expansion [i] based on digital (27, 79, 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]

(30, 30+49, 42)-Net over F₃ — Digital

Digital (30, 79, 42)-net over F₃, using

t-expansion [i] based on digital (29, 79, 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]

(30, 30+49, 121)-Net in Base 3 — Upper bound on s

There is no (30, 79, 122)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁷⁹, 122, S₃, 49), but
- the linear programming bound shows that MÂ â‰¥ 2 696619 084610 776501 344031 056269 137248 595352 418845 365791 957429 212653 / 47840 556342 766214 122108 158565 > 3⁷⁹ [i]