MinT - Best Known (128−85, 128, s)-Nets in Base 5

Best Known (128−85, 128, s)-Nets in Base 5

(128−85, 128, 78)-Net over F₅ — Constructive and digital

Digital (43, 128, 78)-net over F₅, using

t-expansion [i] based on digital (38, 128, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(128−85, 128, 84)-Net over F₅ — Digital

Digital (43, 128, 84)-net over F₅, using

net from sequence [i] based on digital (43, 83)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 43 and N(F) ≥ 84, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(128−85, 128, 505)-Net in Base 5 — Upper bound on s

There is no (43, 128, 506)-net in base 5, because

1 times m-reduction [i] would yield (43, 127, 506)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 59794 554205 506635 897744 278943 966999 045101 287405 465184 296304 693660 539561 038047 437893 218625 > 5¹²⁷ [i]