MinT - Best Known (70, 70+26, s)-Nets in Base 4

Best Known (70, 70+26, s)-Nets in Base 4

(70, 70+26, 384)-Net over F₄ — Constructive and digital

Digital (70, 96, 384)-net over F₄, using

t-expansion [i] based on digital (69, 96, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 32, 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]

(70, 70+26, 450)-Net in Base 4 — Constructive

(70, 96, 450)-net in base 4, using

trace code for nets [i] based on (6, 32, 150)-net in base 64, using
- 3 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
  - base change [i] based on digital (1, 30, 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]

(70, 70+26, 772)-Net over F₄ — Digital

Digital (70, 96, 772)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁶, 772, F₄, 26) (dual of [772, 676, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁹⁶, 1023, F₄, 26) (dual of [1023, 927, 27]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(70, 70+26, 52745)-Net in Base 4 — Upper bound on s

There is no (70, 96, 52746)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6278 642530 143053 989034 700454 796486 772056 905339 705723 917184 > 4⁹⁶ [i]