MinT - Best Known (258−93, 258, s)-Nets in Base 4

Best Known (258−93, 258, s)-Nets in Base 4

(258−93, 258, 200)-Net over F₄ — Constructive and digital

Digital (165, 258, 200)-net over F₄, using

t-expansion [i] based on digital (161, 258, 200)-net over F₄, using
- net from sequence [i] based on digital (161, 199)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 161 and N(F) ≥ 200, using
    - F₇ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(258−93, 258, 208)-Net in Base 4 — Constructive

(165, 258, 208)-net in base 4, using

2 times m-reduction [i] based on (165, 260, 208)-net in base 4, using
- trace code for nets [i] based on (35, 130, 104)-net in base 16, using
  - base change [i] based on digital (9, 104, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(258−93, 258, 522)-Net over F₄ — Digital

Digital (165, 258, 522)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−93, 258, 13823)-Net in Base 4 — Upper bound on s

There is no (165, 258, 13824)-net in base 4, because

1 times m-reduction [i] would yield (165, 257, 13824)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53790 065245 911087 731480 320681 919029 431458 161992 843810 262654 584432 478402 794136 922877 222197 862824 401727 903893 782935 686420 934824 549269 234669 886594 994024 134705 > 4²⁵⁷ [i]