MinT - Best Known (128−83, 128, s)-Nets in Base 5

Best Known (128−83, 128, s)-Nets in Base 5

(128−83, 128, 78)-Net over F₅ — Constructive and digital

Digital (45, 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]

(128−83, 128, 88)-Net over F₅ — Digital

Digital (45, 128, 88)-net over F₅, using

net from sequence [i] based on digital (45, 87)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(128−83, 128, 560)-Net in Base 5 — Upper bound on s

There is no (45, 128, 561)-net in base 5, because

1 times m-reduction [i] would yield (45, 127, 561)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 61153 225369 540826 851660 546428 684676 195703 970092 249114 591742 890741 489758 017435 303230 524325 > 5¹²⁷ [i]