MinT - Best Known (246−85, 246, s)-Nets in Base 4

Best Known (246−85, 246, s)-Nets in Base 4

(246−85, 246, 200)-Net over F₄ — Constructive and digital

Digital (161, 246, 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]

(246−85, 246, 240)-Net in Base 4 — Constructive

(161, 246, 240)-net in base 4, using

4 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]

(246−85, 246, 581)-Net over F₄ — Digital

Digital (161, 246, 581)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−85, 246, 17859)-Net in Base 4 — Upper bound on s

There is no (161, 246, 17860)-net in base 4, because

1 times m-reduction [i] would yield (161, 245, 17860)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3203 394201 285577 017619 545003 744544 403722 489381 211787 482848 474742 439267 446420 456917 770201 214344 128940 213510 642481 714326 762865 124520 221224 452794 405608 > 4²⁴⁵ [i]