MinT - Best Known (128−67, 128, s)-Nets in Base 5

Best Known (128−67, 128, s)-Nets in Base 5

(128−67, 128, 82)-Net over F₅ — Constructive and digital

Digital (61, 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]

(128−67, 128, 120)-Net over F₅ — Digital

Digital (61, 128, 120)-net over F₅, using

net from sequence [i] based on digital (61, 119)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(128−67, 128, 1587)-Net in Base 5 — Upper bound on s

There is no (61, 128, 1588)-net in base 5, because

1 times m-reduction [i] would yield (61, 127, 1588)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 59348 746638 154974 699040 823940 686851 292948 138308 156265 571737 442271 435425 344954 103701 687505 > 5¹²⁷ [i]