MinT - Best Known (259−91, 259, s)-Nets in Base 4

Best Known (259−91, 259, s)-Nets in Base 4

(259−91, 259, 200)-Net over F₄ — Constructive and digital

Digital (168, 259, 200)-net over F₄, using

t-expansion [i] based on digital (161, 259, 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]

(259−91, 259, 240)-Net in Base 4 — Constructive

(168, 259, 240)-net in base 4, using

t-expansion [i] based on (167, 259, 240)-net in base 4, using
- 1 times m-reduction [i] based on (167, 260, 240)-net in base 4, using
  - trace code for nets [i] based on (37, 130, 120)-net in base 16, using
    - base change [i] based on digital (11, 104, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(259−91, 259, 574)-Net over F₄ — Digital

Digital (168, 259, 574)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−91, 259, 16591)-Net in Base 4 — Upper bound on s

There is no (168, 259, 16592)-net in base 4, because

1 times m-reduction [i] would yield (168, 258, 16592)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214577 206777 980260 348891 711628 300921 120794 870744 884244 004785 075074 374670 019980 273026 942029 442678 077163 904923 510970 733671 871076 289415 025436 531962 919997 326822 > 4²⁵⁸ [i]