MinT - Best Known (50−34, 50, s)-Nets in Base 4

Best Known (50−34, 50, s)-Nets in Base 4

(50−34, 50, 33)-Net over F₄ — Constructive and digital

Digital (16, 50, 33)-net over F₄, using

t-expansion [i] based on digital (15, 50, 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]

(50−34, 50, 36)-Net over F₄ — Digital

Digital (16, 50, 36)-net over F₄, using

net from sequence [i] based on digital (16, 35)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 16 and N(F) ≥ 36, using
  - Drinfeld modules of rank 1 [i]

(50−34, 50, 127)-Net in Base 4 — Upper bound on s

There is no (16, 50, 128)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 276349 893271 609524 557067 958377 > 4⁵⁰ [i]
extracting embedded orthogonal array [i] would yield OA(4⁵⁰, 128, S₄, 34), but
- the linear programming bound shows that MÂ â‰¥ 259 179668 828085 226279 928489 912831 484693 129044 766852 890167 105874 843246 416090 051379 200000 / 195 946350 820096 305299 454040 471197 976323 170907 400036 668719 > 4⁵⁰ [i]