MinT - Best Known (67, 67+24, s)-Nets in Base 4

Best Known (67, 67+24, s)-Nets in Base 4

(67, 67+24, 384)-Net over F₄ — Constructive and digital

Digital (67, 91, 384)-net over F₄, using

2 times m-reduction [i] based on digital (67, 93, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 31, 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]

(67, 67+24, 450)-Net in Base 4 — Constructive

(67, 91, 450)-net in base 4, using

4¹ times duplication [i] based on (66, 90, 450)-net in base 4, using
- trace code for nets [i] based on (6, 30, 150)-net in base 64, using
  - 5 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]

(67, 67+24, 860)-Net over F₄ — Digital

Digital (67, 91, 860)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹¹, 860, F₄, 24) (dual of [860, 769, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁹¹, 1023, F₄, 24) (dual of [1023, 932, 25]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {−6,−5,…,17}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(67, 67+24, 64833)-Net in Base 4 — Upper bound on s

There is no (67, 91, 64834)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 130775 721083 744947 446929 987527 981802 999816 943018 727120 > 4⁹¹ [i]