MinT - Best Known (18, 56, s)-Nets in Base 4

Best Known (18, 56, s)-Nets in Base 4

(18, 56, 33)-Net over F₄ — Constructive and digital

Digital (18, 56, 33)-net over F₄, using

t-expansion [i] based on digital (15, 56, 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, 56, 41)-Net over F₄ — Digital

Digital (18, 56, 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, 56, 139)-Net in Base 4 — Upper bound on s

There is no (18, 56, 140)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁶, 140, S₄, 38), but
- the linear programming bound shows that MÂ â‰¥ 33224 510089 193434 121103 982150 072093 135568 373830 671910 070349 424562 458616 197168 496640 000000 / 5 929310 526700 508468 903034 465148 531898 302680 222550 630653 > 4⁵⁶ [i]