MinT - Best Known (256−107, 256, s)-Nets in Base 4

Best Known (256−107, 256, s)-Nets in Base 4

(256−107, 256, 138)-Net over F₄ — Constructive and digital

Digital (149, 256, 138)-net over F₄, using

3 times m-reduction [i] based on digital (149, 259, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 76, 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, 183, 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]

(256−107, 256, 316)-Net over F₄ — Digital

Digital (149, 256, 316)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−107, 256, 5368)-Net in Base 4 — Upper bound on s

There is no (149, 256, 5369)-net in base 4, because

1 times m-reduction [i] would yield (149, 255, 5369)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3363 021020 279025 770390 016110 772520 400720 951412 011439 901919 920549 921361 972672 164404 644140 181586 733128 310285 960339 756723 377563 285323 730412 104803 120772 831392 > 4²⁵⁵ [i]