MinT - Best Known (39−29, 39, s)-Nets in Base 5

Best Known (39−29, 39, s)-Nets in Base 5

(39−29, 39, 26)-Net over F₅ — Constructive and digital

Digital (10, 39, 26)-net over F₅, using

t-expansion [i] based on digital (9, 39, 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]

(39−29, 39, 27)-Net over F₅ — Digital

Digital (10, 39, 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]

(39−29, 39, 108)-Net in Base 5 — Upper bound on s

There is no (10, 39, 109)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5³⁹, 109, S₅, 29), but
- the linear programming bound shows that MÂ â‰¥ 145648 940028 071016 724650 183965 248424 488586 490042 507648 468017 578125 / 77 995853 799136 389818 061242 092117 113337 > 5³⁹ [i]