MinT - Best Known (49, 128, s)-Nets in Base 5

Best Known (49, 128, s)-Nets in Base 5

(49, 128, 82)-Net over F₅ — Constructive and digital

Digital (49, 128, 82)-net over F₅, using

t-expansion [i] based on digital (48, 128, 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]

(49, 128, 96)-Net over F₅ — Digital

Digital (49, 128, 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]

(49, 128, 698)-Net in Base 5 — Upper bound on s

There is no (49, 128, 699)-net in base 5, because

1 times m-reduction [i] would yield (49, 127, 699)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 60090 273025 866987 353939 502935 143222 453210 364143 286432 065012 012935 189687 679553 921941 198485 > 5¹²⁷ [i]