MinT - Best Known (248−85, 248, s)-Nets in Base 4

Best Known (248−85, 248, s)-Nets in Base 4

(248−85, 248, 200)-Net over F₄ — Constructive and digital

Digital (163, 248, 200)-net over F₄, using

t-expansion [i] based on digital (161, 248, 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]

(248−85, 248, 240)-Net in Base 4 — Constructive

(163, 248, 240)-net in base 4, using

t-expansion [i] based on (161, 248, 240)-net in base 4, using
- 2 times m-reduction [i] based on (161, 250, 240)-net in base 4, using
  - trace code for nets [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 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]

(248−85, 248, 602)-Net over F₄ — Digital

Digital (163, 248, 602)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−85, 248, 19080)-Net in Base 4 — Upper bound on s

There is no (163, 248, 19081)-net in base 4, because

1 times m-reduction [i] would yield (163, 247, 19081)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51237 691026 551052 173851 305584 569878 127317 600352 620071 217607 008244 577561 142876 157217 395977 254653 460060 395990 479247 651768 314737 447855 155434 829863 681664 > 4²⁴⁷ [i]