MinT - Best Known (111, 111+51, s)-Nets in Base 4

Best Known (111, 111+51, s)-Nets in Base 4

(111, 111+51, 312)-Net over F₄ — Constructive and digital

Digital (111, 162, 312)-net over F₄, using

trace code for nets [i] based on digital (3, 54, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(111, 111+51, 572)-Net over F₄ — Digital

Digital (111, 162, 572)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+51, 25552)-Net in Base 4 — Upper bound on s

There is no (111, 162, 25553)-net in base 4, because

1 times m-reduction [i] would yield (111, 161, 25553)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 551977 136918 037209 182044 144318 577849 704421 631684 935087 523905 404870 558931 521056 487891 024086 105120 > 4¹⁶¹ [i]