MinT - Best Known (78−48, 78, s)-Nets in Base 3

Best Known (78−48, 78, s)-Nets in Base 3

(78−48, 78, 37)-Net over F₃ — Constructive and digital

Digital (30, 78, 37)-net over F₃, using

t-expansion [i] based on digital (27, 78, 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]

(78−48, 78, 42)-Net over F₃ — Digital

Digital (30, 78, 42)-net over F₃, using

t-expansion [i] based on digital (29, 78, 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]

(78−48, 78, 127)-Net in Base 3 — Upper bound on s

There is no (30, 78, 128)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁷⁸, 128, S₃, 48), but
- the linear programming bound shows that MÂ â‰¥ 6 575490 843517 549614 037804 755790 886921 388297 503372 467014 072409 123799 884455 544386 348644 065807 588730 807152 499714 277823 942029 / 372064 864657 803488 591486 645857 495615 261692 996975 549874 881531 815830 072025 751212 712500 > 3⁷⁸ [i]