MinT - Best Known (86−23, 86, s)-Nets in Base 27

Best Known (86−23, 86, s)-Nets in Base 27

(86−23, 86, 1875)-Net over F₂₇ — Constructive and digital

Digital (63, 86, 1875)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 19, 86)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 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, 13, 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 (44, 67, 1789)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶⁷, 1789, F₂₇, 23, 23) (dual of [(1789, 23), 41080, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(27⁶⁷, 19680, F₂₇, 23) (dual of [19680, 19613, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁶⁷, 19683, F₂₇, 23) (dual of [19683, 19616, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(86−23, 86, 1906)-Net in Base 27 — Constructive

(63, 86, 1906)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (7, 18, 116)-net in base 27, using
  - 2 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
    - base change [i] based on digital (2, 15, 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 (45, 68, 1790)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶⁸, 1790, F₂₇, 23, 23) (dual of [(1790, 23), 41102, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(27⁶⁸, 19691, F₂₇, 23) (dual of [19691, 19623, 24]-code), using
      - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
        linear OA(27⁶⁷, 19684, F₂₇, 23) (dual of [19684, 19617, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(27⁶¹, 19684, F₂₇, 21) (dual of [19684, 19623, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(27¹, 7, F₂₇, 1) (dual of [7, 6, 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]

(86−23, 86, 137166)-Net over F₂₇ — Digital

Digital (63, 86, 137166)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−23, 86, large)-Net in Base 27 — Upper bound on s

There is no (63, 86, large)-net in base 27, because

21 times m-reduction [i] would yield (63, 65, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (86−23, 86, s)-Nets in Base 27

(86−23, 86, 1875)-Net over F27 — Constructive and digital

(86−23, 86, 1906)-Net in Base 27 — Constructive

(86−23, 86, 137166)-Net over F27 — Digital

(86−23, 86, large)-Net in Base 27 — Upper bound on s

(86−23, 86, 1875)-Net over F₂₇ — Constructive and digital

(86−23, 86, 137166)-Net over F₂₇ — Digital