MinT - Best Known (16, 16+42, s)-Nets in Base 4

Best Known (16, 16+42, s)-Nets in Base 4

(16, 16+42, 33)-Net over F₄ — Constructive and digital

Digital (16, 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]

(16, 16+42, 36)-Net over F₄ — Digital

Digital (16, 58, 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]

(16, 16+42, 87)-Net in Base 4 — Upper bound on s

There is no (16, 58, 88)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁸, 88, S₄, 42), but
- the linear programming bound shows that MÂ â‰¥ 158 237175 107336 529531 271038 987459 138072 539269 830656 314178 535424 / 1500 577564 280935 407980 323611 > 4⁵⁸ [i]