MinT - Best Known (51−27, 51, s)-Nets in Base 27

Best Known (51−27, 51, s)-Nets in Base 27

(51−27, 51, 146)-Net over F₂₇ — Constructive and digital

Digital (24, 51, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (7, 34, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(51−27, 51, 172)-Net in Base 27 — Constructive

(24, 51, 172)-net in base 27, using

17 times m-reduction [i] based on (24, 68, 172)-net in base 27, using
- base change [i] based on digital (7, 51, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(51−27, 51, 275)-Net over F₂₇ — Digital

Digital (24, 51, 275)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁵¹, 275, F₂₇, 27) (dual of [275, 224, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁵¹, 364, F₂₇, 27) (dual of [364, 313, 28]-code), using
  - 2 times truncation [i] based on linear OA(27⁵³, 366, F₂₇, 29) (dual of [366, 313, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
      1. linear OA(27⁵³, 365, F₂₇, 29) (dual of [365, 312, 30]-code), using an extension C_e(28) of the narrow-sense BCH-code C(I) with length 364Â | 27²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(27⁵², 365, F₂₇, 28) (dual of [365, 313, 29]-code), using an extension C_e(27) of the narrow-sense BCH-code C(I) with length 364Â | 27²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(27⁰, 1, F₂₇, 0) (dual of [1, 1, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(51−27, 51, 69765)-Net in Base 27 — Upper bound on s

There is no (24, 51, 69766)-net in base 27, because

1 times m-reduction [i] would yield (24, 50, 69766)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 370057 228609 673482 145242 703543 643052 769583 272665 901842 949474 340338 559717 > 27⁵⁰ [i]