MinT - Best Known (75−20, 75, s)-Nets in Base 4

Best Known (75−20, 75, s)-Nets in Base 4

(75−20, 75, 384)-Net over F₄ — Constructive and digital

Digital (55, 75, 384)-net over F₄, using

trace code for nets [i] based on digital (5, 25, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(75−20, 75, 450)-Net in Base 4 — Constructive

(55, 75, 450)-net in base 4, using

trace code for nets [i] based on (5, 25, 150)-net in base 64, using
- 3 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
  - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(75−20, 75, 739)-Net over F₄ — Digital

Digital (55, 75, 739)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁵, 739, F₄, 20) (dual of [739, 664, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷⁵, 1023, F₄, 20) (dual of [1023, 948, 21]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(75−20, 75, 49457)-Net in Base 4 — Upper bound on s

There is no (55, 75, 49458)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1427 258934 420903 366900 282242 019082 907697 968920 > 4⁷⁵ [i]