MinT - Best Known (58−13, 58, s)-Nets in Base 27

Best Known (58−13, 58, s)-Nets in Base 27

(58−13, 58, 88630)-Net over F₂₇ — Constructive and digital

Digital (45, 58, 88630)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 56)-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 (0, 6, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (36, 49, 88574)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁴⁹, 88574, F₂₇, 13, 13) (dual of [(88574, 13), 1151413, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁴⁹, 531445, F₂₇, 13) (dual of [531445, 531396, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(27⁴⁹, 531441, F₂₇, 13) (dual of [531441, 531392, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(27⁴⁵, 531441, F₂₇, 12) (dual of [531441, 531396, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [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]

(58−13, 58, 88657)-Net in Base 27 — Constructive

(45, 58, 88657)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (2, 8, 82)-net in base 27, using
  - base change [i] based on digital (0, 6, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
2. digital (37, 50, 88575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁵⁰, 88575, F₂₇, 13, 13) (dual of [(88575, 13), 1151425, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁵⁰, 531451, F₂₇, 13) (dual of [531451, 531401, 14]-code), using
      - constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(27⁴⁹, 531442, F₂₇, 13) (dual of [531442, 531393, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 27⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(27⁴¹, 531442, F₂₇, 11) (dual of [531442, 531401, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 27⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (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]

(58−13, 58, 1685186)-Net over F₂₇ — Digital

Digital (45, 58, 1685186)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58−13, 58, large)-Net in Base 27 — Upper bound on s

There is no (45, 58, large)-net in base 27, because

11 times m-reduction [i] would yield (45, 47, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (58−13, 58, s)-Nets in Base 27

(58−13, 58, 88630)-Net over F27 — Constructive and digital

(58−13, 58, 88657)-Net in Base 27 — Constructive

(58−13, 58, 1685186)-Net over F27 — Digital

(58−13, 58, large)-Net in Base 27 — Upper bound on s

(58−13, 58, 88630)-Net over F₂₇ — Constructive and digital

(58−13, 58, 1685186)-Net over F₂₇ — Digital