MinT - Best Known (68, 92, s)-Nets in Base 4

Best Known (68, 92, s)-Nets in Base 4

(68, 92, 384)-Net over F₄ — Constructive and digital

Digital (68, 92, 384)-net over F₄, using

t-expansion [i] based on digital (67, 92, 384)-net over F₄, using
- 1 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]

(68, 92, 450)-Net in Base 4 — Constructive

(68, 92, 450)-net in base 4, using

1 times m-reduction [i] based on (68, 93, 450)-net in base 4, using
- trace code for nets [i] based on (6, 31, 150)-net in base 64, using
  - 4 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]

(68, 92, 917)-Net over F₄ — Digital

Digital (68, 92, 917)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹², 917, F₄, 24) (dual of [917, 825, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁹², 1035, F₄, 24) (dual of [1035, 943, 25]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
    1. linear OA(4⁹¹, 1024, F₄, 25) (dual of [1024, 933, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(4⁸¹, 1024, F₄, 22) (dual of [1024, 943, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(4¹, 11, F₄, 1) (dual of [11, 10, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(68, 92, 72774)-Net in Base 4 — Upper bound on s

There is no (68, 92, 72775)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 24 523637 782555 129091 635663 954130 387921 118654 873169 173016 > 4⁹² [i]