MinT - Best Known (21, 21+19, s)-Nets in Base 27

Best Known (21, 21+19, s)-Nets in Base 27

(21, 21+19, 152)-Net over F₂₇ — Constructive and digital

Digital (21, 40, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 15, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (6, 25, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(21, 21+19, 200)-Net in Base 27 — Constructive

(21, 40, 200)-net in base 27, using

base change [i] based on digital (11, 30, 200)-net over F₈₁, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 10, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
  2. digital (1, 20, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)

(21, 21+19, 523)-Net over F₂₇ — Digital

Digital (21, 40, 523)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴⁰, 523, F₂₇, 19) (dual of [523, 483, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁴⁰, 741, F₂₇, 19) (dual of [741, 701, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(27³⁷, 730, F₂₇, 19) (dual of [730, 693, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(27²⁹, 730, F₂₇, 15) (dual of [730, 701, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    3. linear OA(27³, 11, F₂₇, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
      - discarding factors / shortening the dual code based on linear OA(27³, 27, F₂₇, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
        Reedâ€“Solomon code RS(24,27) [i]

(21, 21+19, 254300)-Net in Base 27 — Upper bound on s

There is no (21, 40, 254301)-net in base 27, because

1 times m-reduction [i] would yield (21, 39, 254301)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 66 557424 918932 008474 368499 570272 431793 325405 584208 515347 > 27³⁹ [i]