MinT - Best Known (157−54, 157, s)-Nets in Base 4

Best Known (157−54, 157, s)-Nets in Base 4

(157−54, 157, 157)-Net over F₄ — Constructive and digital

Digital (103, 157, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 37, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (66, 120, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 60, 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]

(157−54, 157, 196)-Net in Base 4 — Constructive

(103, 157, 196)-net in base 4, using

3 times m-reduction [i] based on (103, 160, 196)-net in base 4, using
- trace code for nets [i] based on (23, 80, 98)-net in base 16, using
  - base change [i] based on digital (7, 64, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(157−54, 157, 399)-Net over F₄ — Digital

Digital (103, 157, 399)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−54, 157, 11516)-Net in Base 4 — Upper bound on s

There is no (103, 157, 11517)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 33405 299795 217638 737513 144324 635700 151293 966937 714012 411264 489038 003503 691384 551320 870060 168020 > 4¹⁵⁷ [i]