MinT - Best Known (232−127, 232, s)-Nets in Base 4

Best Known (232−127, 232, s)-Nets in Base 4

(232−127, 232, 130)-Net over F₄ — Constructive and digital

Digital (105, 232, 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]

(232−127, 232, 144)-Net over F₄ — Digital

Digital (105, 232, 144)-net over F₄, using

t-expansion [i] based on digital (91, 232, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(232−127, 232, 1255)-Net in Base 4 — Upper bound on s

There is no (105, 232, 1256)-net in base 4, because

1 times m-reduction [i] would yield (105, 231, 1256)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 345165 462753 552491 955692 209721 328754 334917 072659 120729 886499 302025 873759 654921 647629 180485 087415 710960 702528 122038 861929 987085 243671 463640 > 4²³¹ [i]