MinT - Best Known (138, 138+107, s)-Nets in Base 4

Best Known (138, 138+107, s)-Nets in Base 4

(138, 138+107, 132)-Net over F₄ — Constructive and digital

Digital (138, 245, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 65, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 180, 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]

(138, 138+107, 264)-Net over F₄ — Digital

Digital (138, 245, 264)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+107, 4015)-Net in Base 4 — Upper bound on s

There is no (138, 245, 4016)-net in base 4, because

1 times m-reduction [i] would yield (138, 244, 4016)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 804 413287 397801 413225 448289 065512 944256 348220 195391 345152 879065 198289 421497 405572 962615 299690 697767 354035 997843 255542 646391 687372 910171 193266 063144 > 4²⁴⁴ [i]