MinT - Best Known (34, 34+19, s)-Nets in Base 27

Best Known (34, 34+19, s)-Nets in Base 27

(34, 34+19, 336)-Net over F₂₇ — Constructive and digital

Digital (34, 53, 336)-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, 3, 28)-net over F₂₇ (see above)
9. digital (0, 4, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
10. digital (0, 6, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
11. digital (0, 9, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
12. digital (0, 19, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(34, 34+19, 729)-Net in Base 27 — Constructive

(34, 53, 729)-net in base 27, using

27¹ times duplication [i] based on (33, 52, 729)-net in base 27, using
- base change [i] based on digital (20, 39, 729)-net over F₈₁, using
  - 81² times duplication [i] based on digital (18, 37, 729)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81³⁷, 729, F₈₁, 19, 19) (dual of [(729, 19), 13814, 20]-NRT-code), using
      - OOA 9-folding and stacking with additional row [i] based on linear OA(81³⁷, 6562, F₈₁, 19) (dual of [6562, 6525, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(34, 34+19, 4770)-Net over F₂₇ — Digital

Digital (34, 53, 4770)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 34+19, large)-Net in Base 27 — Upper bound on s

There is no (34, 53, large)-net in base 27, because

17 times m-reduction [i] would yield (34, 36, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]