MinT - Best Known (232−155, 232, s)-Nets in Base 4

Best Known (232−155, 232, s)-Nets in Base 4

(232−155, 232, 104)-Net over F₄ — Constructive and digital

Digital (77, 232, 104)-net over F₄, using

t-expansion [i] based on digital (73, 232, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(232−155, 232, 112)-Net over F₄ — Digital

Digital (77, 232, 112)-net over F₄, using

t-expansion [i] based on digital (73, 232, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(232−155, 232, 567)-Net in Base 4 — Upper bound on s

There is no (77, 232, 568)-net in base 4, because

1 times m-reduction [i] would yield (77, 231, 568)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 308237 061700 976969 977918 989671 455789 482716 400835 611691 180624 090514 399911 322613 541557 590358 284943 993127 396128 582046 255523 249375 462214 200080 > 4²³¹ [i]