MinT - Best Known (77−20, 77, s)-Nets in Base 4

Best Known (77−20, 77, s)-Nets in Base 4

(77−20, 77, 384)-Net over F₄ — Constructive and digital

Digital (57, 77, 384)-net over F₄, using

1 times m-reduction [i] based on digital (57, 78, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 26, 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]

(77−20, 77, 450)-Net in Base 4 — Constructive

(57, 77, 450)-net in base 4, using

1 times m-reduction [i] based on (57, 78, 450)-net in base 4, using
- trace code for nets [i] based on (5, 26, 150)-net in base 64, using
  - 2 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]

(77−20, 77, 864)-Net over F₄ — Digital

Digital (57, 77, 864)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁷, 864, F₄, 20) (dual of [864, 787, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷⁷, 1035, F₄, 20) (dual of [1035, 958, 21]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [i] based on
    1. linear OA(4⁷⁶, 1024, F₄, 21) (dual of [1024, 948, 22]-code), using an extension C_e(20) of 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]
    2. linear OA(4⁶⁶, 1024, F₄, 18) (dual of [1024, 958, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(77−20, 77, 65262)-Net in Base 4 — Upper bound on s

There is no (57, 77, 65263)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 22837 105846 632962 633440 749575 282326 248291 781573 > 4⁷⁷ [i]