MinT - Best Known (25, 25+27, s)-Nets in Base 9

Best Known (25, 25+27, s)-Nets in Base 9

(25, 25+27, 78)-Net over F₉ — Constructive and digital

Digital (25, 52, 78)-net over F₉, using

t-expansion [i] based on digital (22, 52, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(25, 25+27, 82)-Net in Base 9 — Constructive

(25, 52, 82)-net in base 9, using

2 times m-reduction [i] based on (25, 54, 82)-net in base 9, using
- base change [i] based on digital (7, 36, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(25, 25+27, 103)-Net over F₉ — Digital

Digital (25, 52, 103)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 25+27, 3917)-Net in Base 9 — Upper bound on s

There is no (25, 52, 3918)-net in base 9, because

1 times m-reduction [i] would yield (25, 51, 3918)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 641918 123087 951094 951643 615945 553978 059193 034545 > 9⁵¹ [i]