MinT - Best Known (159, 159+97, s)-Nets in Base 4

Best Known (159, 159+97, s)-Nets in Base 4

(159, 159+97, 160)-Net over F₄ — Constructive and digital

Digital (159, 256, 160)-net over F₄, using

t-expansion [i] based on digital (157, 256, 160)-net over F₄, using
- 3 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 175, 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]

(159, 159+97, 436)-Net over F₄ — Digital

Digital (159, 256, 436)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+97, 9825)-Net in Base 4 — Upper bound on s

There is no (159, 256, 9826)-net in base 4, because

1 times m-reduction [i] would yield (159, 255, 9826)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3353 517892 647355 278783 404238 245053 476244 555223 706280 048161 884640 686278 279089 311235 594282 173256 740476 326556 865483 300357 593676 999731 530066 674488 477188 128572 > 4²⁵⁵ [i]