MinT - Best Known (37−14, 37, s)-Nets in Base 27

Best Known (37−14, 37, s)-Nets in Base 27

(37−14, 37, 280)-Net over F₂₇ — Constructive and digital

Digital (23, 37, 280)-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, 1, 28)-net over F₂₇ (see above)
3. digital (0, 1, 28)-net over F₂₇ (see above)
4. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
5. digital (0, 2, 28)-net over F₂₇ (see above)
6. digital (0, 2, 28)-net over F₂₇ (see above)
7. 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]
8. digital (0, 4, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
9. digital (0, 7, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
10. digital (0, 14, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(37−14, 37, 937)-Net in Base 27 — Constructive

(23, 37, 937)-net in base 27, using

27¹ times duplication [i] based on (22, 36, 937)-net in base 27, using
- base change [i] based on digital (13, 27, 937)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²⁷, 937, F₈₁, 14, 14) (dual of [(937, 14), 13091, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(81²⁷, 6559, F₈₁, 14) (dual of [6559, 6532, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(81²⁷, 6561, F₈₁, 14) (dual of [6561, 6534, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(37−14, 37, 2591)-Net over F₂₇ — Digital

Digital (23, 37, 2591)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(37−14, 37, 4783270)-Net in Base 27 — Upper bound on s

There is no (23, 37, 4783271)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 91297 710178 490258 474828 581765 072096 188224 393901 639339 > 27³⁷ [i]