MinT - Best Known (249−87, 249, s)-Nets in Base 4

Best Known (249−87, 249, s)-Nets in Base 4

(249−87, 249, 200)-Net over F₄ — Constructive and digital

Digital (162, 249, 200)-net over F₄, using

t-expansion [i] based on digital (161, 249, 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]

(249−87, 249, 240)-Net in Base 4 — Constructive

(162, 249, 240)-net in base 4, using

t-expansion [i] based on (161, 249, 240)-net in base 4, using
- 1 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]

(249−87, 249, 565)-Net over F₄ — Digital

Digital (162, 249, 565)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−87, 249, 16663)-Net in Base 4 — Upper bound on s

There is no (162, 249, 16664)-net in base 4, because

1 times m-reduction [i] would yield (162, 248, 16664)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204690 664303 366437 666771 530574 745047 708574 574939 931556 593288 178249 430071 825115 124713 021548 297242 000867 017670 776847 420733 831262 589909 346398 945451 698160 > 4²⁴⁸ [i]