MinT - Best Known (36−17, 36, s)-Nets in Base 27

Best Known (36−17, 36, s)-Nets in Base 27

(36−17, 36, 146)-Net over F₂₇ — Constructive and digital

Digital (19, 36, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 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, 24, 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]

(36−17, 36, 200)-Net in Base 27 — Constructive

(19, 36, 200)-net in base 27, using

base change [i] based on digital (10, 27, 200)-net over F₈₁, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 9, 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, 18, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)

(36−17, 36, 534)-Net over F₂₇ — Digital

Digital (19, 36, 534)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27³⁶, 534, F₂₇, 17) (dual of [534, 498, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(27³⁶, 741, F₂₇, 17) (dual of [741, 705, 18]-code), using
  - constructionÂ X applied to C([0,8]) âŠ‚ C([0,6]) [i] based on
    1. 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]
    2. linear OA(27²⁵, 730, F₂₇, 13) (dual of [730, 705, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (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]

(36−17, 36, 264805)-Net in Base 27 — Upper bound on s

There is no (19, 36, 264806)-net in base 27, because

1 times m-reduction [i] would yield (19, 35, 264806)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 125 238768 220997 631397 666065 711151 709182 901535 458769 > 27³⁵ [i]