MinT - Best Known (66−47, 66, s)-Nets in Base 5

Best Known (66−47, 66, s)-Nets in Base 5

(66−47, 66, 43)-Net over F₅ — Constructive and digital

Digital (19, 66, 43)-net over F₅, using

t-expansion [i] based on digital (18, 66, 43)-net over F₅, using
- net from sequence [i] based on digital (18, 42)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 17, N(F)Â =Â 42, and 1 place with degree 2 [i] based on function field F/F₅ with g(F) = 17 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(66−47, 66, 45)-Net over F₅ — Digital

Digital (19, 66, 45)-net over F₅, using

net from sequence [i] based on digital (19, 44)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 19 and N(F) ≥ 45, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(66−47, 66, 194)-Net in Base 5 — Upper bound on s

There is no (19, 66, 195)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁶⁶, 195, S₅, 47), but
- the linear programming bound shows that MÂ â‰¥ 5778 642374 961171 005858 384950 541644 608646 561762 096995 896264 437475 895090 940551 943785 976618 528366 088867 187500 000000 000000 000000 / 410745 131436 451894 158641 306360 451078 631309 149686 982683 533600 888107 150686 828531 > 5⁶⁶ [i]