MinT - Best Known (85, 85+6, s)-Nets in Base 27

Best Known (85, 85+6, s)-Nets in Base 27

(85, 85+6, large)-Net over F₂₇ — Constructive and digital

Digital (85, 91, large)-net over F₂₇, using

t-expansion [i] based on digital (80, 91, 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, 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]
  18. digital (0, 1, 265722)-net over F₂₇ (see above)
  19. digital (0, 1, 265722)-net over F₂₇ (see above)
  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 (3, 5, 265722)-net over F₂₇, using
    - s-reduction based on digital (3, 5, 551881)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
  24. digital (3, 5, 265722)-net over F₂₇ (see above)
  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 (12, 17, 265722)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27¹⁷, 265722, F₂₇, 5, 5) (dual of [(265722, 5), 1328593, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(27¹⁷, 531445, F₂₇, 5) (dual of [531445, 531428, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(27¹⁷, 531441, F₂₇, 5) (dual of [531441, 531424, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        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(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 (40, 51, 1677720)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(85, 85+6, large)-Net in Base 27 — Upper bound on s

There is no (85, 91, large)-net in base 27, because

4 times m-reduction [i] would yield (85, 87, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (85, 85+6, s)-Nets in Base 27

(85, 85+6, large)-Net over F27 — Constructive and digital

(85, 85+6, large)-Net in Base 27 — Upper bound on s

(85, 85+6, large)-Net over F₂₇ — Constructive and digital