MinT - Best Known (73−45, 73, s)-Nets in Base 3

Best Known (73−45, 73, s)-Nets in Base 3

(73−45, 73, 37)-Net over F₃ — Constructive and digital

Digital (28, 73, 37)-net over F₃, using

t-expansion [i] based on digital (27, 73, 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]

(73−45, 73, 39)-Net over F₃ — Digital

Digital (28, 73, 39)-net over F₃, using

t-expansion [i] based on digital (27, 73, 39)-net over F₃, using
- net from sequence [i] based on digital (27, 38)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 27 and N(F) ≥ 39, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(73−45, 73, 122)-Net in Base 3 — Upper bound on s

There is no (28, 73, 123)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁷³, 123, S₃, 45), but
- the linear programming bound shows that MÂ â‰¥ 2808 640754 709561 673413 288749 360382 305600 098849 480190 178164 374813 402246 542737 575036 629671 056464 419225 742014 266460 475560 838182 553344 275311 / 35976 248284 132713 371830 997095 816817 658282 486367 564623 621887 133661 233734 765511 477094 428331 280913 134080 > 3⁷³ [i]