MinT - Best Known (145−77, 145, s)-Nets in Base 4

Best Known (145−77, 145, s)-Nets in Base 4

(145−77, 145, 66)-Net over F₄ — Constructive and digital

Digital (68, 145, 66)-net over F₄, using

t-expansion [i] based on digital (49, 145, 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]

(145−77, 145, 99)-Net over F₄ — Digital

Digital (68, 145, 99)-net over F₄, using

t-expansion [i] based on digital (61, 145, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(145−77, 145, 926)-Net in Base 4 — Upper bound on s

There is no (68, 145, 927)-net in base 4, because

1 times m-reduction [i] would yield (68, 144, 927)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 501 487442 454049 403196 328224 467546 657991 645590 647812 352115 872518 525345 804125 953433 913700 > 4¹⁴⁴ [i]