MinT - Best Known (40−30, 40, s)-Nets in Base 5

Best Known (40−30, 40, s)-Nets in Base 5

(40−30, 40, 26)-Net over F₅ — Constructive and digital

Digital (10, 40, 26)-net over F₅, using

t-expansion [i] based on digital (9, 40, 26)-net over F₅, using
- net from sequence [i] based on digital (9, 25)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 9 and N(F) ≥ 26, using
    - an explicitly constructive algebraic function field [i]

(40−30, 40, 27)-Net over F₅ — Digital

Digital (10, 40, 27)-net over F₅, using

net from sequence [i] based on digital (10, 26)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 10 and N(F) ≥ 27, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(40−30, 40, 106)-Net in Base 5 — Upper bound on s

There is no (10, 40, 107)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 9114 229293 184050 457118 705845 > 5⁴⁰ [i]
extracting embedded orthogonal array [i] would yield OA(5⁴⁰, 107, S₅, 30), but
- the linear programming bound shows that MÂ â‰¥ 2 444583 551382 201850 781520 395853 708391 699789 897762 821055 948734 283447 265625 / 251 181963 463860 852519 484964 340757 927650 764452 > 5⁴⁰ [i]