MinT - Best Known (178−113, 178, s)-Nets in Base 4

Best Known (178−113, 178, s)-Nets in Base 4

(178−113, 178, 66)-Net over F₄ — Constructive and digital

Digital (65, 178, 66)-net over F₄, using

t-expansion [i] based on digital (49, 178, 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]

(178−113, 178, 99)-Net over F₄ — Digital

Digital (65, 178, 99)-net over F₄, using

t-expansion [i] based on digital (61, 178, 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]

(178−113, 178, 533)-Net in Base 4 — Upper bound on s

There is no (65, 178, 534)-net in base 4, because

1 times m-reduction [i] would yield (65, 177, 534)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 37082 465174 723958 197074 246547 258993 883169 742935 889377 938919 034193 856907 102122 672178 313523 456882 262889 666880 > 4¹⁷⁷ [i]