MinT - Best Known (80−12, 80, s)-Nets in Base 27

Best Known (80−12, 80, s)-Nets in Base 27

(80−12, 80, 1575276)-Net over F₂₇ — Constructive and digital

Digital (68, 80, 1575276)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (18, 24, 177176)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. digital (15, 21, 177148)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27²¹, 177148, F₂₇, 6, 6) (dual of [(177148, 6), 1062867, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(27²¹, 531444, F₂₇, 6) (dual of [531444, 531423, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(27²¹, 531445, F₂₇, 6) (dual of [531445, 531424, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(27²¹, 531441, F₂₇, 6) (dual of [531441, 531420, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        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(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]
2. digital (44, 56, 1398100)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁵⁶, 1398100, F₂₇, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(27⁵⁶, 8388600, F₂₇, 12) (dual of [8388600, 8388544, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁵⁶, large, F₂₇, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(80−12, 80, large)-Net over F₂₇ — Digital

Digital (68, 80, large)-net over F₂₇, using

6 times m-reduction [i] based on digital (68, 86, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁸⁶, large, F₂₇, 18) (dual of [large, large−86, 19]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(80−12, 80, large)-Net in Base 27 — Upper bound on s

There is no (68, 80, large)-net in base 27, because

10 times m-reduction [i] would yield (68, 70, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (80−12, 80, s)-Nets in Base 27

(80−12, 80, 1575276)-Net over F27 — Constructive and digital

(80−12, 80, large)-Net over F27 — Digital

(80−12, 80, large)-Net in Base 27 — Upper bound on s

(80−12, 80, 1575276)-Net over F₂₇ — Constructive and digital

(80−12, 80, large)-Net over F₂₇ — Digital