MinT - Best Known (63, 63+81, s)-Nets in Base 4

Best Known (63, 63+81, s)-Nets in Base 4

(63, 63+81, 66)-Net over F₄ — Constructive and digital

Digital (63, 144, 66)-net over F₄, using

t-expansion [i] based on digital (49, 144, 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]

(63, 63+81, 99)-Net over F₄ — Digital

Digital (63, 144, 99)-net over F₄, using

t-expansion [i] based on digital (61, 144, 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]

(63, 63+81, 714)-Net in Base 4 — Upper bound on s

There is no (63, 144, 715)-net in base 4, because

1 times m-reduction [i] would yield (63, 143, 715)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 129 669704 567857 045607 857128 009553 138435 637832 564521 228554 779663 919814 189015 577073 347490 > 4¹⁴³ [i]