MinT - Best Known (163−97, 163, s)-Nets in Base 4

Best Known (163−97, 163, s)-Nets in Base 4

(163−97, 163, 66)-Net over F₄ — Constructive and digital

Digital (66, 163, 66)-net over F₄, using

t-expansion [i] based on digital (49, 163, 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]

(163−97, 163, 99)-Net over F₄ — Digital

Digital (66, 163, 99)-net over F₄, using

t-expansion [i] based on digital (61, 163, 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]

(163−97, 163, 633)-Net in Base 4 — Upper bound on s

There is no (66, 163, 634)-net in base 4, because

1 times m-reduction [i] would yield (66, 162, 634)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34 789208 326044 851994 721824 154148 083974 483874 453020 969553 417014 091835 863236 681764 339665 607882 117737 > 4¹⁶² [i]