MinT - Best Known (198−137, 198, s)-Nets in Base 4

Best Known (198−137, 198, s)-Nets in Base 4

(198−137, 198, 66)-Net over F₄ — Constructive and digital

Digital (61, 198, 66)-net over F₄, using

t-expansion [i] based on digital (49, 198, 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]

(198−137, 198, 99)-Net over F₄ — Digital

Digital (61, 198, 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]

(198−137, 198, 429)-Net in Base 4 — Upper bound on s

There is no (61, 198, 430)-net in base 4, because

1 times m-reduction [i] would yield (61, 197, 430)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40570 560476 681725 011677 901648 551083 084536 747040 464925 018952 903152 440817 445344 966269 795842 404891 174261 835143 936028 809762 > 4¹⁹⁷ [i]