MinT - Best Known (216−107, 216, s)-Nets in Base 4

Best Known (216−107, 216, s)-Nets in Base 4

(216−107, 216, 130)-Net over F₄ — Constructive and digital

Digital (109, 216, 130)-net over F₄, using

t-expansion [i] based on digital (105, 216, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(216−107, 216, 165)-Net over F₄ — Digital

Digital (109, 216, 165)-net over F₄, using

net from sequence [i] based on digital (109, 164)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
  - Drinfeld modules of rank 1 [i]

(216−107, 216, 1857)-Net in Base 4 — Upper bound on s

There is no (109, 216, 1858)-net in base 4, because

1 times m-reduction [i] would yield (109, 215, 1858)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2792 786588 511190 084731 019121 180190 415741 534645 225224 827880 570936 986591 950630 699622 565350 682433 634947 325062 754514 819284 260333 136560 > 4²¹⁵ [i]