MinT - Best Known (135−85, 135, s)-Nets in Base 5

Best Known (135−85, 135, s)-Nets in Base 5

(135−85, 135, 82)-Net over F₅ — Constructive and digital

Digital (50, 135, 82)-net over F₅, using

t-expansion [i] based on digital (48, 135, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(135−85, 135, 96)-Net over F₅ — Digital

Digital (50, 135, 96)-net over F₅, using

t-expansion [i] based on digital (49, 135, 96)-net over F₅, using
- net from sequence [i] based on digital (49, 95)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 49 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(135−85, 135, 670)-Net in Base 5 — Upper bound on s

There is no (50, 135, 671)-net in base 5, because

1 times m-reduction [i] would yield (50, 134, 671)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4705 246259 342566 830527 191895 769272 343986 890068 394474 688688 037108 089116 519409 680623 453548 767065 > 5¹³⁴ [i]