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

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

(27, 27+23, 170)-Net over F₂₇ — Constructive and digital

Digital (27, 50, 170)-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 (9, 32, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(27, 27+23, 224)-Net in Base 27 — Constructive

(27, 50, 224)-net in base 27, using

6 times m-reduction [i] based on (27, 56, 224)-net in base 27, using
- base change [i] based on digital (13, 42, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(27, 27+23, 720)-Net over F₂₇ — Digital

Digital (27, 50, 720)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁵⁰, 720, F₂₇, 23) (dual of [720, 670, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁵⁰, 747, F₂₇, 23) (dual of [747, 697, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,8]) [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(27³³, 730, F₂₇, 17) (dual of [730, 697, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
    3. linear OA(27⁵, 17, F₂₇, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,27)), using
      - discarding factors / shortening the dual code based on linear OA(27⁵, 27, F₂₇, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
        Reedâ€“Solomon code RS(22,27) [i]

(27, 27+23, 448859)-Net in Base 27 — Upper bound on s

There is no (27, 50, 448860)-net in base 27, because

1 times m-reduction [i] would yield (27, 49, 448860)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13703 414158 023515 974361 082311 193100 987683 271570 308619 302701 696455 539953 > 27⁴⁹ [i]