MinT - Best Known (84−64, 84, s)-Nets in Base 5

Best Known (84−64, 84, s)-Nets in Base 5

(84−64, 84, 43)-Net over F₅ — Constructive and digital

Digital (20, 84, 43)-net over F₅, using

t-expansion [i] based on digital (18, 84, 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]

(84−64, 84, 45)-Net over F₅ — Digital

Digital (20, 84, 45)-net over F₅, using

t-expansion [i] based on digital (19, 84, 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]

(84−64, 84, 130)-Net in Base 5 — Upper bound on s

There is no (20, 84, 131)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁸⁴, 131, S₅, 64), but
- the linear programming bound shows that MÂ â‰¥ 5001 544234 538435 915919 627195 800020 646445 476478 107105 879542 499978 919290 612379 595917 236230 206981 417723 000049 591064 453125 / 85369 053210 566041 966253 484563 756858 333751 172542 180225 401184 > 5⁸⁴ [i]