MinT - Best Known (75−46, 75, s)-Nets in Base 3

Best Known (75−46, 75, s)-Nets in Base 3

(75−46, 75, 37)-Net over F₃ — Constructive and digital

Digital (29, 75, 37)-net over F₃, using

t-expansion [i] based on digital (27, 75, 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]

(75−46, 75, 42)-Net over F₃ — Digital

Digital (29, 75, 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]

(75−46, 75, 127)-Net in Base 3 — Upper bound on s

There is no (29, 75, 128)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁷⁵, 128, S₃, 46), but
- the linear programming bound shows that MÂ â‰¥ 70509 034008 273048 497949 012415 256823 095516 948639 518464 642288 291654 944885 730085 391182 130810 751743 773920 008127 422894 926894 729600 054599 / 107409 815338 072251 071920 212049 180487 610108 565772 602900 313365 362537 798880 614700 874814 997458 059264 > 3⁷⁵ [i]