MinT - Best Known (106−81, 106, s)-Nets in Base 5

Best Known (106−81, 106, s)-Nets in Base 5

(106−81, 106, 51)-Net over F₅ — Constructive and digital

Digital (25, 106, 51)-net over F₅, using

t-expansion [i] based on digital (22, 106, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(106−81, 106, 55)-Net over F₅ — Digital

Digital (25, 106, 55)-net over F₅, using

t-expansion [i] based on digital (23, 106, 55)-net over F₅, using
- net from sequence [i] based on digital (23, 54)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(106−81, 106, 139)-Net in Base 5 — Upper bound on s

There is no (25, 106, 140)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5¹⁰⁶, 140, S₅, 81), but
- the linear programming bound shows that MÂ â‰¥ 6 593241 895990 744825 682423 763191 499653 100382 657555 024729 830188 585932 848610 582368 564791 977405 548095 703125 / 46159 159144 205061 204938 029056 > 5¹⁰⁶ [i]