MinT - Best Known (109−16, 109, s)-Nets in Base 27

Best Known (109−16, 109, s)-Nets in Base 27

(109−16, 109, 1181464)-Net over F₂₇ — Constructive and digital

Digital (93, 109, 1181464)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (25, 33, 132889)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. digital (21, 29, 132861)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27²⁹, 132861, F₂₇, 8, 8) (dual of [(132861, 8), 1062859, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(27²⁹, 531444, F₂₇, 8) (dual of [531444, 531415, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(27²⁹, 531445, F₂₇, 8) (dual of [531445, 531416, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(27²⁹, 531441, F₂₇, 8) (dual of [531441, 531412, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(27²⁵, 531441, F₂₇, 7) (dual of [531441, 531416, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [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 (60, 76, 1048575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷⁶, 1048575, F₂₇, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(27⁷⁶, 8388600, F₂₇, 16) (dual of [8388600, 8388524, 17]-code), using
      - discarding factors / shortening the dual code 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]

(109−16, 109, large)-Net over F₂₇ — Digital

Digital (93, 109, large)-net over F₂₇, using

27³ times duplication [i] based on digital (90, 106, large)-net over F₂₇, using
- t-expansion [i] based on digital (84, 106, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁰⁶, large, F₂₇, 22) (dual of [large, large−106, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(109−16, 109, large)-Net in Base 27 — Upper bound on s

There is no (93, 109, large)-net in base 27, because

14 times m-reduction [i] would yield (93, 95, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (109−16, 109, s)-Nets in Base 27

(109−16, 109, 1181464)-Net over F27 — Constructive and digital

(109−16, 109, large)-Net over F27 — Digital

(109−16, 109, large)-Net in Base 27 — Upper bound on s

(109−16, 109, 1181464)-Net over F₂₇ — Constructive and digital

(109−16, 109, large)-Net over F₂₇ — Digital