MinT - Best Known (220−75, 220, s)-Nets in Base 4

Best Known (220−75, 220, s)-Nets in Base 4

(220−75, 220, 164)-Net over F₄ — Constructive and digital

Digital (145, 220, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 58, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (87, 162, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 81, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(220−75, 220, 240)-Net in Base 4 — Constructive

(145, 220, 240)-net in base 4, using

t-expansion [i] based on (143, 220, 240)-net in base 4, using
- trace code for nets [i] based on (33, 110, 120)-net in base 16, using
  - base change [i] based on digital (11, 88, 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]

(220−75, 220, 551)-Net over F₄ — Digital

Digital (145, 220, 551)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−75, 220, 17848)-Net in Base 4 — Upper bound on s

There is no (145, 220, 17849)-net in base 4, because

1 times m-reduction [i] would yield (145, 219, 17849)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 710772 650772 665580 632682 840352 293166 410722 358743 669524 770271 204538 688197 122947 555567 020148 893069 701803 948315 966924 344100 079419 536320 > 4²¹⁹ [i]