MinT - Best Known (66−46, 66, s)-Nets in Base 5

Best Known (66−46, 66, s)-Nets in Base 5

(66−46, 66, 43)-Net over F₅ — Constructive and digital

Digital (20, 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]

(66−46, 66, 45)-Net over F₅ — Digital

Digital (20, 66, 45)-net over F₅, using

t-expansion [i] based on 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]

(66−46, 66, 217)-Net in Base 5 — Upper bound on s

There is no (20, 66, 218)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁶⁶, 218, S₅, 46), but
- the linear programming bound shows that MÂ â‰¥ 4 697609 786040 779276 505227 505722 373609 044135 841255 862093 171119 134248 239080 966537 585482 001304 626464 843750 000000 000000 / 341 366350 358561 713362 034506 460187 187286 087053 467825 060609 177888 774643 > 5⁶⁶ [i]