MinT - Best Known (224−79, 224, s)-Nets in Base 4

Best Known (224−79, 224, s)-Nets in Base 4

(224−79, 224, 163)-Net over F₄ — Constructive and digital

Digital (145, 224, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 54, 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 (91, 170, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 85, 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]

(224−79, 224, 208)-Net in Base 4 — Constructive

(145, 224, 208)-net in base 4, using

2 times m-reduction [i] based on (145, 226, 208)-net in base 4, using
- trace code for nets [i] based on (32, 113, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (32, 115, 104)-net in base 16, using
    - base change [i] based on digital (9, 92, 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]

(224−79, 224, 498)-Net over F₄ — Digital

Digital (145, 224, 498)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−79, 224, 14186)-Net in Base 4 — Upper bound on s

There is no (145, 224, 14187)-net in base 4, because

1 times m-reduction [i] would yield (145, 223, 14187)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 971810 492787 516439 071243 501601 238938 623093 610243 396338 125526 309234 228705 614273 918883 379031 260831 183779 962132 399056 059399 110712 384480 > 4²²³ [i]