MinT - Best Known (248−145, 248, s)-Nets in Base 4

Best Known (248−145, 248, s)-Nets in Base 4

(248−145, 248, 104)-Net over F₄ — Constructive and digital

Digital (103, 248, 104)-net over F₄, using

t-expansion [i] based on digital (73, 248, 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]

(248−145, 248, 144)-Net over F₄ — Digital

Digital (103, 248, 144)-net over F₄, using

t-expansion [i] based on digital (91, 248, 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]

(248−145, 248, 1012)-Net in Base 4 — Upper bound on s

There is no (103, 248, 1013)-net in base 4, because

1 times m-reduction [i] would yield (103, 247, 1013)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52310 719290 929817 423485 245674 193511 808966 483225 280337 414211 469043 410296 291960 030094 258353 366701 736870 384274 904657 619629 761852 779569 557350 390909 209398 > 4²⁴⁷ [i]