MinT - Best Known (229−123, 229, s)-Nets in Base 4

Best Known (229−123, 229, s)-Nets in Base 4

(229−123, 229, 130)-Net over F₄ — Constructive and digital

Digital (106, 229, 130)-net over F₄, using

t-expansion [i] based on digital (105, 229, 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]

(229−123, 229, 144)-Net over F₄ — Digital

Digital (106, 229, 144)-net over F₄, using

t-expansion [i] based on digital (91, 229, 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]

(229−123, 229, 1348)-Net in Base 4 — Upper bound on s

There is no (106, 229, 1349)-net in base 4, because

1 times m-reduction [i] would yield (106, 228, 1349)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 193741 621566 335150 879782 688541 847886 152856 502765 142662 540146 774592 291856 280056 770425 910277 020845 307492 431586 382304 899698 140975 295587 932000 > 4²²⁸ [i]