MinT - Best Known (111−53, 111, s)-Nets in Base 4

Best Known (111−53, 111, s)-Nets in Base 4

(111−53, 111, 66)-Net over F₄ — Constructive and digital

Digital (58, 111, 66)-net over F₄, using

t-expansion [i] based on digital (49, 111, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(111−53, 111, 101)-Net over F₄ — Digital

Digital (58, 111, 101)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−53, 111, 1218)-Net in Base 4 — Upper bound on s

There is no (58, 111, 1219)-net in base 4, because

1 times m-reduction [i] would yield (58, 110, 1219)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 693398 081900 581617 500056 576598 954106 488338 635196 983292 914909 055224 > 4¹¹⁰ [i]