MinT - Best Known (109−83, 109, s)-Nets in Base 5

Best Known (109−83, 109, s)-Nets in Base 5

(109−83, 109, 51)-Net over F₅ — Constructive and digital

Digital (26, 109, 51)-net over F₅, using

t-expansion [i] based on digital (22, 109, 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]

(109−83, 109, 55)-Net over F₅ — Digital

Digital (26, 109, 55)-net over F₅, using

t-expansion [i] based on digital (23, 109, 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]

(109−83, 109, 146)-Net in Base 5 — Upper bound on s

There is no (26, 109, 147)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5¹⁰⁹, 147, S₅, 83), but
- the linear programming bound shows that MÂ â‰¥ 97832 005097 623965 961965 091455 398251 491670 818930 622769 438002 739592 061328 226246 796901 932611 945085 227489 471435 546875 / 5 803319 642355 292721 815985 390811 721467 > 5¹⁰⁹ [i]