MinT - Best Known (89−19, 89, s)-Nets in Base 27

Best Known (89−19, 89, s)-Nets in Base 27

(89−19, 89, 59135)-Net over F₂₇ — Constructive and digital

Digital (70, 89, 59135)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 16, 86)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 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 (2, 11, 48)-net over F₂₇, using
      - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]
2. digital (54, 73, 59049)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷³, 59049, F₂₇, 19, 19) (dual of [(59049, 19), 1121858, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(27⁷³, 531442, F₂₇, 19) (dual of [531442, 531369, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 531442Â | 27⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(89−19, 89, 59166)-Net in Base 27 — Constructive

(70, 89, 59166)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (6, 15, 116)-net in base 27, using
  - 1 times m-reduction [i] based on (6, 16, 116)-net in base 27, using
    - base change [i] based on digital (2, 12, 116)-net over F₈₁, using
      - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]
2. digital (55, 74, 59050)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷⁴, 59050, F₂₇, 19, 19) (dual of [(59050, 19), 1121876, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(27⁷⁴, 531451, F₂₇, 19) (dual of [531451, 531377, 20]-code), using
      - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
        linear OA(27⁷³, 531442, F₂₇, 19) (dual of [531442, 531369, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 27⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(27⁶⁵, 531442, F₂₇, 17) (dual of [531442, 531377, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 27⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(27¹, 9, F₂₇, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(89−19, 89, 3471015)-Net over F₂₇ — Digital

Digital (70, 89, 3471015)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−19, 89, large)-Net in Base 27 — Upper bound on s

There is no (70, 89, large)-net in base 27, because

17 times m-reduction [i] would yield (70, 72, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (89−19, 89, s)-Nets in Base 27

(89−19, 89, 59135)-Net over F27 — Constructive and digital

(89−19, 89, 59166)-Net in Base 27 — Constructive

(89−19, 89, 3471015)-Net over F27 — Digital

(89−19, 89, large)-Net in Base 27 — Upper bound on s

(89−19, 89, 59135)-Net over F₂₇ — Constructive and digital

(89−19, 89, 3471015)-Net over F₂₇ — Digital