MinT - Best Known (29, 138, s)-Nets in Base 5

Best Known (29, 138, s)-Nets in Base 5

(29, 138, 51)-Net over F₅ — Constructive and digital

Digital (29, 138, 51)-net over F₅, using

t-expansion [i] based on digital (22, 138, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(29, 138, 56)-Net over F₅ — Digital

Digital (29, 138, 56)-net over F₅, using

net from sequence [i] based on digital (29, 55)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 29 and N(F) ≥ 56, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(29, 138, 146)-Net in Base 5 — Upper bound on s

There is no (29, 138, 147)-net in base 5, because

7 times m-reduction [i] would yield (29, 131, 147)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹³¹, 147, S₅, 102), but
  - the linear programming bound shows that MÂ â‰¥ 249 236477 010921 693845 684281 987701 342304 847343 669838 056342 020625 628930 208250 721989 315934 479236 602783 203125 / 6 326951 585981 > 5¹³¹ [i]