MinT - Best Known (18, 18+40, s)-Nets in Base 4

Best Known (18, 18+40, s)-Nets in Base 4

(18, 18+40, 33)-Net over F₄ — Constructive and digital

Digital (18, 58, 33)-net over F₄, using

t-expansion [i] based on digital (15, 58, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(18, 18+40, 41)-Net over F₄ — Digital

Digital (18, 58, 41)-net over F₄, using

net from sequence [i] based on digital (18, 40)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
  - Drinfeld modules of rank 1 [i]

(18, 18+40, 129)-Net in Base 4 — Upper bound on s

There is no (18, 58, 130)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁸, 130, S₄, 40), but
- the linear programming bound shows that MÂ â‰¥ 209544 936299 479389 121518 912044 089753 196453 227215 118049 279760 432327 769939 564811 820879 196123 613172 070974 880613 527842 029138 321043 777447 471353 954304 / 2 470482 844369 173135 750828 733014 583913 640615 187243 459992 006489 938479 554355 225141 791107 956885 358375 430667 629325 > 4⁵⁸ [i]