MinT - Best Known (248−103, 248, s)-Nets in Base 4

Best Known (248−103, 248, s)-Nets in Base 4

(248−103, 248, 138)-Net over F₄ — Constructive and digital

Digital (145, 248, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 72, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 176, 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]

(248−103, 248, 314)-Net over F₄ — Digital

Digital (145, 248, 314)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−103, 248, 5410)-Net in Base 4 — Upper bound on s

There is no (145, 248, 5411)-net in base 4, because

1 times m-reduction [i] would yield (145, 247, 5411)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51339 142222 305538 760760 146811 953449 587940 966211 545759 667183 093054 775829 352940 148869 289660 583794 583027 157306 299582 845465 965695 822728 174340 462258 687680 > 4²⁴⁷ [i]