MinT - Best Known (116−69, 116, s)-Nets in Base 5

Best Known (116−69, 116, s)-Nets in Base 5

(116−69, 116, 78)-Net over F₅ — Constructive and digital

Digital (47, 116, 78)-net over F₅, using

t-expansion [i] based on digital (38, 116, 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]

(116−69, 116, 92)-Net over F₅ — Digital

Digital (47, 116, 92)-net over F₅, using

net from sequence [i] based on digital (47, 91)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 47 and N(F) ≥ 92, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(116−69, 116, 757)-Net in Base 5 — Upper bound on s

There is no (47, 116, 758)-net in base 5, because

1 times m-reduction [i] would yield (47, 115, 758)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 241 254045 020406 388036 683240 240219 032780 572246 033589 252892 867548 985499 456546 538625 > 5¹¹⁵ [i]