MinT - Best Known (256−157, 256, s)-Nets in Base 4

Best Known (256−157, 256, s)-Nets in Base 4

(256−157, 256, 104)-Net over F₄ — Constructive and digital

Digital (99, 256, 104)-net over F₄, using

t-expansion [i] based on digital (73, 256, 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]

(256−157, 256, 144)-Net over F₄ — Digital

Digital (99, 256, 144)-net over F₄, using

t-expansion [i] based on digital (91, 256, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(256−157, 256, 862)-Net in Base 4 — Upper bound on s

There is no (99, 256, 863)-net in base 4, because

1 times m-reduction [i] would yield (99, 255, 863)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3560 252801 248988 450859 174351 769899 365057 935585 528939 821738 364555 864993 535851 210977 423715 217165 389263 718223 212452 900513 378549 704640 059158 100992 207194 528235 > 4²⁵⁵ [i]