MinT - Best Known (24−11, 24, s)-Nets in Base 27

Best Known (24−11, 24, s)-Nets in Base 27

(24−11, 24, 168)-Net over F₂₇ — Constructive and digital

Digital (13, 24, 168)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 28)-net over F₂₇, using
  - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
3. digital (0, 2, 28)-net over F₂₇ (see above)
4. digital (0, 3, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
5. digital (0, 5, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
6. digital (0, 11, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(24−11, 24, 200)-Net in Base 27 — Constructive

(13, 24, 200)-net in base 27, using

base change [i] based on digital (7, 18, 200)-net over F₈₁, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 6, 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, 12, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)

(24−11, 24, 722)-Net over F₂₇ — Digital

Digital (13, 24, 722)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27²⁴, 722, F₂₇, 11) (dual of [722, 698, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(27²⁴, 741, F₂₇, 11) (dual of [741, 717, 12]-code), using
  - constructionÂ X applied to C([0,5]) âŠ‚ C([0,3]) [i] based on
    1. linear OA(27²¹, 730, F₂₇, 11) (dual of [730, 709, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
    2. linear OA(27¹³, 730, F₂₇, 7) (dual of [730, 717, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (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]

(24−11, 24, 384710)-Net in Base 27 — Upper bound on s

There is no (13, 24, 384711)-net in base 27, because

1 times m-reduction [i] would yield (13, 23, 384711)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 834 393455 668364 020561 468267 472399 > 27²³ [i]