MinT - Best Known (28, 28+23, s)-Nets in Base 27

Best Known (28, 28+23, s)-Nets in Base 27

(28, 28+23, 176)-Net over F₂₇ — Constructive and digital

Digital (28, 51, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 18, 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]
2. digital (10, 33, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(28, 28+23, 232)-Net in Base 27 — Constructive

(28, 51, 232)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (4, 15, 82)-net in base 27, using
  - 1 times m-reduction [i] based on (4, 16, 82)-net in base 27, using
    - base change [i] based on digital (0, 12, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
2. (13, 36, 150)-net in base 27, using
  - base change [i] based on digital (4, 27, 150)-net over F₈₁, using
    - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(28, 28+23, 790)-Net over F₂₇ — Digital

Digital (28, 51, 790)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁵¹, 790, F₂₇, 23) (dual of [790, 739, 24]-code), using
- 50 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 11 times 0, 1, 33 times 0) [i] based on linear OA(27⁴⁶, 735, F₂₇, 23) (dual of [735, 689, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(27⁴⁵, 730, F₂₇, 23) (dual of [730, 685, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(27⁴¹, 730, F₂₇, 21) (dual of [730, 689, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    3. linear OA(27¹, 5, F₂₇, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(28, 28+23, 605669)-Net in Base 27 — Upper bound on s

There is no (28, 51, 605670)-net in base 27, because

1 times m-reduction [i] would yield (28, 50, 605670)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 369990 501037 521944 079562 613391 466990 849759 234060 176130 351727 257483 948449 > 27⁵⁰ [i]