MinT - Best Known (18, 18+36, s)-Nets in Base 4

Best Known (18, 18+36, s)-Nets in Base 4

(18, 18+36, 33)-Net over F₄ — Constructive and digital

Digital (18, 54, 33)-net over F₄, using

t-expansion [i] based on digital (15, 54, 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+36, 41)-Net over F₄ — Digital

Digital (18, 54, 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+36, 147)-Net in Base 4 — Upper bound on s

There is no (18, 54, 148)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 358 102505 146457 237257 814308 817995 > 4⁵⁴ [i]
extracting embedded orthogonal array [i] would yield OA(4⁵⁴, 148, S₄, 36), but
- the linear programming bound shows that MÂ â‰¥ 360 104803 308131 437253 911585 010681 444562 627305 886362 911504 773250 289965 509640 192000 / 1 033080 609162 065310 926475 587063 900861 412950 368521 > 4⁵⁴ [i]