MinT - Best Known (219−113, 219, s)-Nets in Base 4

Best Known (219−113, 219, s)-Nets in Base 4

(219−113, 219, 130)-Net over F₄ — Constructive and digital

Digital (106, 219, 130)-net over F₄, using

t-expansion [i] based on digital (105, 219, 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]

(219−113, 219, 144)-Net over F₄ — Digital

Digital (106, 219, 144)-net over F₄, using

t-expansion [i] based on digital (91, 219, 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]

(219−113, 219, 1551)-Net in Base 4 — Upper bound on s

There is no (106, 219, 1552)-net in base 4, because

1 times m-reduction [i] would yield (106, 218, 1552)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182857 056905 153781 006560 589371 641181 318970 573124 570076 765218 965541 415056 968335 523326 197828 891669 993561 702493 823759 706445 954496 272224 > 4²¹⁸ [i]