MinT - Best Known (237−84, 237, s)-Nets in Base 4

Best Known (237−84, 237, s)-Nets in Base 4

(237−84, 237, 163)-Net over F₄ — Constructive and digital

Digital (153, 237, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 57, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (96, 180, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 90, 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]

(237−84, 237, 208)-Net in Base 4 — Constructive

(153, 237, 208)-net in base 4, using

3 times m-reduction [i] based on (153, 240, 208)-net in base 4, using
- trace code for nets [i] based on (33, 120, 104)-net in base 16, using
  - base change [i] based on digital (9, 96, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(237−84, 237, 515)-Net over F₄ — Digital

Digital (153, 237, 515)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−84, 237, 13706)-Net in Base 4 — Upper bound on s

There is no (153, 237, 13707)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48843 238787 641931 702460 549357 097758 191606 549012 687051 287368 424302 749357 954340 817546 641978 289363 313580 096601 261498 367541 977784 206590 428936 359140 > 4²³⁷ [i]