MinT - Best Known (137, 137+105, s)-Nets in Base 4

Best Known (137, 137+105, s)-Nets in Base 4

(137, 137+105, 132)-Net over F₄ — Constructive and digital

Digital (137, 242, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 64, 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, 178, 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]

(137, 137+105, 266)-Net over F₄ — Digital

Digital (137, 242, 266)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+105, 4117)-Net in Base 4 — Upper bound on s

There is no (137, 242, 4118)-net in base 4, because

1 times m-reduction [i] would yield (137, 241, 4118)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 592815 834219 021715 899479 957671 290831 970973 585157 781539 937672 656222 659317 285416 756178 472416 741415 638406 089099 828008 707376 977708 466715 530328 317568 > 4²⁴¹ [i]