MinT - Best Known (257−93, 257, s)-Nets in Base 4

Best Known (257−93, 257, s)-Nets in Base 4

(257−93, 257, 200)-Net over F₄ — Constructive and digital

Digital (164, 257, 200)-net over F₄, using

t-expansion [i] based on digital (161, 257, 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]

(257−93, 257, 208)-Net in Base 4 — Constructive

(164, 257, 208)-net in base 4, using

1 times m-reduction [i] based on (164, 258, 208)-net in base 4, using
- trace code for nets [i] based on (35, 129, 104)-net in base 16, using
  - 1 times m-reduction [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]

(257−93, 257, 513)-Net over F₄ — Digital

Digital (164, 257, 513)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−93, 257, 13411)-Net in Base 4 — Upper bound on s

There is no (164, 257, 13412)-net in base 4, because

1 times m-reduction [i] would yield (164, 256, 13412)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13425 334292 807071 210187 281565 601618 139956 801577 902212 010056 421498 197070 555771 891015 101087 605985 429326 897926 926057 942453 108692 683949 839430 582241 746328 494384 > 4²⁵⁶ [i]