MinT - Best Known (20−8, 20, s)-Nets in Base 27

Best Known (20−8, 20, s)-Nets in Base 27

(20−8, 20, 756)-Net over F₂₇ — Constructive and digital

Digital (12, 20, 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, 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]
21. digital (0, 1, 28)-net over F₂₇ (see above)
22. digital (0, 1, 28)-net over F₂₇ (see above)
23. digital (0, 1, 28)-net over F₂₇ (see above)
24. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
25. digital (0, 2, 28)-net over F₂₇ (see above)
26. digital (0, 4, 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, 8, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(20−8, 20, 1610)-Net over F₂₇ — Digital

Digital (12, 20, 1610)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27²⁰, 1610, F₂₇, 8) (dual of [1610, 1590, 9]-code), using
- 873 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 11 times 0, 1, 64 times 0, 1, 241 times 0, 1, 553 times 0) [i] based on linear OA(27¹⁵, 732, F₂₇, 8) (dual of [732, 717, 9]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,5]), C₂Â =Â C([0,6]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,5]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,6]) [i] based on
    1. 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]
    2. linear OA(27¹³, 728, F₂₇, 7) (dual of [728, 715, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(27¹⁵, 728, F₂₇, 8) (dual of [728, 713, 9]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,6}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    4. 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]
    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)

(20−8, 20, 1640)-Net in Base 27 — Constructive

(12, 20, 1640)-net in base 27, using

base change [i] based on digital (7, 15, 1640)-net over F₈₁, using
- net defined by OOA [i] based on linear OOA(81¹⁵, 1640, F₈₁, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
  - OA 4-folding and stacking [i] based on linear OA(81¹⁵, 6560, F₈₁, 8) (dual of [6560, 6545, 9]-code), using
    - discarding factors / shortening the dual code based on linear OA(81¹⁵, 6561, F₈₁, 8) (dual of [6561, 6546, 9]-code), using
      - an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(20−8, 20, 2982)-Net in Base 27

(12, 20, 2982)-net in base 27, using

base change [i] based on digital (7, 15, 2982)-net over F₈₁, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81¹⁵, 2982, F₈₁, 2, 8) (dual of [(2982, 2), 5949, 9]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81¹⁵, 3281, F₈₁, 2, 8) (dual of [(3281, 2), 6547, 9]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(81¹⁵, 6562, F₈₁, 8) (dual of [6562, 6547, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(81¹⁵, 6563, F₈₁, 8) (dual of [6563, 6548, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(81¹⁵, 6561, F₈₁, 8) (dual of [6561, 6546, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(81¹³, 6561, F₈₁, 7) (dual of [6561, 6548, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(81⁰, 2, F₈₁, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(20−8, 20, 1221511)-Net in Base 27 — Upper bound on s

There is no (12, 20, 1221512)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 42391 174886 371921 090622 536513 > 27²⁰ [i]