MinT - Best Known (65−45, 65, s)-Nets in Base 5

Best Known (65−45, 65, s)-Nets in Base 5

(65−45, 65, 43)-Net over F₅ — Constructive and digital

Digital (20, 65, 43)-net over F₅, using

t-expansion [i] based on digital (18, 65, 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]

(65−45, 65, 45)-Net over F₅ — Digital

Digital (20, 65, 45)-net over F₅, using

t-expansion [i] based on digital (19, 65, 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]

(65−45, 65, 225)-Net in Base 5 — Upper bound on s

There is no (20, 65, 226)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁶⁵, 226, S₅, 45), but
- the linear programming bound shows that MÂ â‰¥ 2 261990 824381 937477 904834 249569 576308 343383 908716 127843 663808 516652 125035 761855 542659 759521 484375 000000 / 788 649499 436360 668671 839494 491009 208630 003248 162377 694287 > 5⁶⁵ [i]