MinT - Best Known (240−83, 240, s)-Nets in Base 4

Best Known (240−83, 240, s)-Nets in Base 4

(240−83, 240, 164)-Net over F₄ — Constructive and digital

Digital (157, 240, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 62, 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 (95, 178, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 89, 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]

(240−83, 240, 240)-Net in Base 4 — Constructive

(157, 240, 240)-net in base 4, using

t-expansion [i] based on (155, 240, 240)-net in base 4, using
- trace code for nets [i] based on (35, 120, 120)-net in base 16, using
  - base change [i] based on digital (11, 96, 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]

(240−83, 240, 566)-Net over F₄ — Digital

Digital (157, 240, 566)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−83, 240, 17358)-Net in Base 4 — Upper bound on s

There is no (157, 240, 17359)-net in base 4, because

1 times m-reduction [i] would yield (157, 239, 17359)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 780782 543859 573383 123229 685204 809845 100361 308596 284950 045781 636921 822477 659269 452710 873064 861613 862568 539750 303684 626341 594151 651228 462735 905444 > 4²³⁹ [i]