MinT - Best Known (76−13, 76, s)-Nets in Base 27

Best Known (76−13, 76, s)-Nets in Base 27

(76−13, 76, 1398866)-Net over F₂₇ — Constructive and digital

Digital (63, 76, 1398866)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 15, 766)-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, 0, 28)-net over F₂₇ (see above)
    22. 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]
    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 (1, 7, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (48, 61, 1398100)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶¹, 1398100, F₂₇, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁶¹, 8388601, F₂₇, 13) (dual of [8388601, 8388540, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(76−13, 76, 1400287)-Net in Base 27 — Constructive

(63, 76, 1400287)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (9, 15, 2187)-net in base 27, using
  - net defined by OOA [i] based on OOA(27¹⁵, 2187, S₂₇, 6, 6), using
    - OA 3-folding and stacking [i] based on OA(27¹⁵, 6561, S₂₇, 6), using
      - discarding factors based on OA(27¹⁵, 6563, S₂₇, 6), using
        discarding parts of the base [i] based on linear OA(81¹¹, 6563, F₈₁, 6) (dual of [6563, 6552, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(81¹¹, 6561, F₈₁, 6) (dual of [6561, 6550, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(81⁹, 6561, F₈₁, 5) (dual of [6561, 6552, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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]
2. digital (48, 61, 1398100)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶¹, 1398100, F₂₇, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁶¹, 8388601, F₂₇, 13) (dual of [8388601, 8388540, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(76−13, 76, large)-Net over F₂₇ — Digital

Digital (63, 76, large)-net over F₂₇, using

t-expansion [i] based on digital (60, 76, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁷⁶, large, F₂₇, 16) (dual of [large, large−76, 17]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(76−13, 76, large)-Net in Base 27 — Upper bound on s

There is no (63, 76, large)-net in base 27, because

11 times m-reduction [i] would yield (63, 65, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (76−13, 76, s)-Nets in Base 27

(76−13, 76, 1398866)-Net over F27 — Constructive and digital

(76−13, 76, 1400287)-Net in Base 27 — Constructive

(76−13, 76, large)-Net over F27 — Digital

(76−13, 76, large)-Net in Base 27 — Upper bound on s

(76−13, 76, 1398866)-Net over F₂₇ — Constructive and digital

(76−13, 76, large)-Net over F₂₇ — Digital