MinT - Best Known (258−111, 258, s)-Nets in Base 4

Best Known (258−111, 258, s)-Nets in Base 4

(258−111, 258, 137)-Net over F₄ — Constructive and digital

Digital (147, 258, 137)-net over F₄, using

t-expansion [i] based on digital (145, 258, 137)-net over F₄, using
- 1 times m-reduction [i] based on digital (145, 259, 137)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 72, 33)-net over F₄, using
      - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
    2. digital (73, 187, 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]

(258−111, 258, 290)-Net over F₄ — Digital

Digital (147, 258, 290)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−111, 258, 4581)-Net in Base 4 — Upper bound on s

There is no (147, 258, 4582)-net in base 4, because

1 times m-reduction [i] would yield (147, 257, 4582)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53741 436518 904028 099363 549333 204998 435510 875876 712185 186115 760745 466365 681297 519045 843930 496679 704210 738346 490059 824369 651210 956309 581485 790605 025514 027680 > 4²⁵⁷ [i]