MinT - Best Known (71−13, 71, s)-Nets in Base 27

Best Known (71−13, 71, s)-Nets in Base 27

(71−13, 71, 1398166)-Net over F₂₇ — Constructive and digital

Digital (58, 71, 1398166)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 66)-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 (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]

(71−13, 71, 1398200)-Net in Base 27 — Constructive

(58, 71, 1398200)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (4, 10, 100)-net in base 27, using
  - 2 times m-reduction [i] based on (4, 12, 100)-net in base 27, using
    - base change [i] based on digital (1, 9, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, 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]

(71−13, 71, large)-Net over F₂₇ — Digital

Digital (58, 71, large)-net over F₂₇, using

t-expansion [i] based on digital (56, 71, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁷¹, large, F₂₇, 15) (dual of [large, large−71, 16]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(71−13, 71, large)-Net in Base 27 — Upper bound on s

There is no (58, 71, large)-net in base 27, because

11 times m-reduction [i] would yield (58, 60, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (71−13, 71, s)-Nets in Base 27

(71−13, 71, 1398166)-Net over F27 — Constructive and digital

(71−13, 71, 1398200)-Net in Base 27 — Constructive

(71−13, 71, large)-Net over F27 — Digital

(71−13, 71, large)-Net in Base 27 — Upper bound on s

(71−13, 71, 1398166)-Net over F₂₇ — Constructive and digital

(71−13, 71, large)-Net over F₂₇ — Digital