MinT - Best Known (71−9, 71, s)-Nets in Base 27

Best Known (71−9, 71, s)-Nets in Base 27

(71−9, 71, large)-Net over F₂₇ — Constructive and digital

Digital (62, 71, large)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 265722)-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, 265722)-net over F₂₇ (see above)
3. digital (0, 0, 265722)-net over F₂₇ (see above)
4. digital (0, 0, 265722)-net over F₂₇ (see above)
5. digital (0, 0, 265722)-net over F₂₇ (see above)
6. digital (0, 0, 265722)-net over F₂₇ (see above)
7. digital (0, 0, 265722)-net over F₂₇ (see above)
8. digital (0, 0, 265722)-net over F₂₇ (see above)
9. digital (0, 0, 265722)-net over F₂₇ (see above)
10. digital (0, 0, 265722)-net over F₂₇ (see above)
11. digital (0, 0, 265722)-net over F₂₇ (see above)
12. digital (0, 0, 265722)-net over F₂₇ (see above)
13. digital (0, 0, 265722)-net over F₂₇ (see above)
14. digital (0, 0, 265722)-net over F₂₇ (see above)
15. digital (0, 0, 265722)-net over F₂₇ (see above)
16. digital (0, 0, 265722)-net over F₂₇ (see above)
17. digital (0, 0, 265722)-net over F₂₇ (see above)
18. digital (0, 0, 265722)-net over F₂₇ (see above)
19. digital (0, 1, 265722)-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]
20. digital (0, 1, 265722)-net over F₂₇ (see above)
21. digital (0, 1, 265722)-net over F₂₇ (see above)
22. digital (0, 1, 265722)-net over F₂₇ (see above)
23. digital (0, 1, 265722)-net over F₂₇ (see above)
24. digital (3, 5, 265722)-net over F₂₇, using
  - s-reduction based on digital (3, 5, 551881)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
25. digital (4, 7, 265722)-net over F₂₇, using
  - s-reduction based on digital (4, 7, 532899)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁷, 532899, F₂₇, 3, 3) (dual of [(532899, 3), 1598690, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(27⁷, 532899, F₂₇, 2, 3) (dual of [(532899, 2), 1065791, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]
26. digital (9, 13, 265722)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27¹³, 265722, F₂₇, 4, 4) (dual of [(265722, 4), 1062875, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(27¹³, 531444, F₂₇, 4) (dual of [531444, 531431, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(27¹³, 531445, F₂₇, 4) (dual of [531445, 531432, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(27¹³, 531441, F₂₇, 4) (dual of [531441, 531428, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(27⁹, 531441, F₂₇, 3) (dual of [531441, 531432, 4]-code or 531441-cap in PG(8,27)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(27⁰, 4, F₂₇, 0) (dual of [4, 4, 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]
27. digital (32, 41, 2097150)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁴¹, 2097150, F₂₇, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(27⁴¹, 8388601, F₂₇, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁴¹, large, F₂₇, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(71−9, 71, large)-Net in Base 27 — Upper bound on s

There is no (62, 71, large)-net in base 27, because

7 times m-reduction [i] would yield (62, 64, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (71−9, 71, s)-Nets in Base 27

(71−9, 71, large)-Net over F27 — Constructive and digital

(71−9, 71, large)-Net in Base 27 — Upper bound on s

(71−9, 71, large)-Net over F₂₇ — Constructive and digital