MinT - Best Known (16−7, 16, s)-Nets in Base 27

Best Known (16−7, 16, s)-Nets in Base 27

(16−7, 16, 756)-Net over F₂₇ — Constructive and digital

Digital (9, 16, 756)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 28)-net over F₂₇, using
  - s-reduction based on digital (0, 0, s)-net over F₂₇ with arbitrarily large s, using
    - arbitrary set of 27⁰ points [i]
2. digital (0, 0, 28)-net over F₂₇ (see above)
3. digital (0, 0, 28)-net over F₂₇ (see above)
4. digital (0, 0, 28)-net over F₂₇ (see above)
5. digital (0, 0, 28)-net over F₂₇ (see above)
6. digital (0, 0, 28)-net over F₂₇ (see above)
7. digital (0, 0, 28)-net over F₂₇ (see above)
8. digital (0, 0, 28)-net over F₂₇ (see above)
9. digital (0, 0, 28)-net over F₂₇ (see above)
10. digital (0, 0, 28)-net over F₂₇ (see above)
11. digital (0, 0, 28)-net over F₂₇ (see above)
12. digital (0, 0, 28)-net over F₂₇ (see above)
13. digital (0, 0, 28)-net over F₂₇ (see above)
14. digital (0, 0, 28)-net over F₂₇ (see above)
15. digital (0, 0, 28)-net over F₂₇ (see above)
16. digital (0, 0, 28)-net over F₂₇ (see above)
17. digital (0, 0, 28)-net over F₂₇ (see above)
18. digital (0, 0, 28)-net over F₂₇ (see above)
19. digital (0, 0, 28)-net over F₂₇ (see above)
20. digital (0, 0, 28)-net over F₂₇ (see above)
21. 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]
22. digital (0, 1, 28)-net over F₂₇ (see above)
23. digital (0, 1, 28)-net over F₂₇ (see above)
24. digital (0, 1, 28)-net over F₂₇ (see above)
25. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
26. 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]
27. digital (0, 7, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(16−7, 16, 893)-Net over F₂₇ — Digital

Digital (9, 16, 893)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁶, 893, F₂₇, 7) (dual of [893, 877, 8]-code), using
- 158 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 18 times 0, 1, 138 times 0) [i] based on linear OA(27¹³, 732, F₂₇, 7) (dual of [732, 719, 8]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,4]), C₂Â =Â C([0,5]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,4]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,5]) [i] based on
    1. linear OA(27¹¹, 728, F₂₇, 6) (dual of [728, 717, 7]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,4}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    2. linear OA(27¹¹, 728, F₂₇, 6) (dual of [728, 717, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    3. linear OA(27¹³, 728, F₂₇, 7) (dual of [728, 715, 8]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,5}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    4. linear OA(27⁹, 728, F₂₇, 5) (dual of [728, 719, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
    5. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
    6. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code) (see above)

(16−7, 16, 1002833)-Net in Base 27 — Upper bound on s

There is no (9, 16, 1002834)-net in base 27, because

1 times m-reduction [i] would yield (9, 15, 1002834)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2954 319030 966707 756353 > 27¹⁵ [i]

Best Known (16−7, 16, s)-Nets in Base 27

(16−7, 16, 756)-Net over F27 — Constructive and digital

(16−7, 16, 893)-Net over F27 — Digital

(16−7, 16, 1002833)-Net in Base 27 — Upper bound on s

(16−7, 16, 756)-Net over F₂₇ — Constructive and digital

(16−7, 16, 893)-Net over F₂₇ — Digital