MinT - Best Known (65, 65+99, s)-Nets in Base 4

Best Known (65, 65+99, s)-Nets in Base 4

(65, 65+99, 66)-Net over F₄ — Constructive and digital

Digital (65, 164, 66)-net over F₄, using

t-expansion [i] based on digital (49, 164, 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]

(65, 65+99, 99)-Net over F₄ — Digital

Digital (65, 164, 99)-net over F₄, using

t-expansion [i] based on digital (61, 164, 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]

(65, 65+99, 601)-Net in Base 4 — Upper bound on s

There is no (65, 164, 602)-net in base 4, because

1 times m-reduction [i] would yield (65, 163, 602)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 139 299928 402575 181935 214497 244783 272488 572683 065327 501509 680023 308328 573600 582263 169496 669121 480474 > 4¹⁶³ [i]