MinT - Best Known (13, 13+∞, s)-Nets in Base 5

Best Known (13, 13+∞, s)-Nets in Base 5

(13, 13+∞, 34)-Net over F₅ — Constructive and digital

Digital (13, m, 34)-net over F₅ for arbitrarily large m, using

net from sequence [i] based on digital (13, 33)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 11, N(F)Â =Â 32, and 2 places with degree 2 [i] based on function field F/F₅ with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(13, 13+∞, 36)-Net over F₅ — Digital

Digital (13, m, 36)-net over F₅ for arbitrarily large m, using

net from sequence [i] based on digital (13, 35)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 13 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(13, 13+∞, 65)-Net in Base 5 — Upper bound on s

There is no (13, m, 66)-net in base 5 for arbitrarily large m, because

m-reduction [i] would yield (13, 194, 66)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5¹⁹⁴, 66, S₅, 3, 181), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 458013 924450 580029 645803 114358 217144 107442 559617 877900 996242 182848 521834 185004 221271 138338 982052 522208 032740 991257 014684 379100 799560 546875 / 91 > 5¹⁹⁴ [i]