MinT - Best Known (37−16, 37, s)-Nets in Base 27

Best Known (37−16, 37, s)-Nets in Base 27

(37−16, 37, 164)-Net over F₂₇ — Constructive and digital

Digital (21, 37, 164)-net over F₂₇, using

1 times m-reduction [i] based on digital (21, 38, 164)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 14, 82)-net over F₂₇, using
    - s-reduction based on digital (6, 14, 84)-net over F₂₇, using
      - generalized (u,Â u+v)-construction [i] based on
        digital (0, 2, 28)-net over F₂₇, using
        kÂ =Â 2 construction [i]
        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]
        digital (0, 8, 28)-net over F₂₇, using
        net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
  2. digital (7, 24, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
        a function field by SÃ©mirat [i]

(37−16, 37, 216)-Net in Base 27 — Constructive

(21, 37, 216)-net in base 27, using

1 times m-reduction [i] based on (21, 38, 216)-net in base 27, using
- (u,Â u+v)-construction [i] based on
  1. (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. (9, 26, 116)-net in base 27, using
    - 2 times m-reduction [i] based on (9, 28, 116)-net in base 27, using
      - base change [i] based on digital (2, 21, 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]

(37−16, 37, 885)-Net over F₂₇ — Digital

Digital (21, 37, 885)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27³⁷, 885, F₂₇, 16) (dual of [885, 848, 17]-code), using
- 147 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 12 times 0, 1, 36 times 0, 1, 92 times 0) [i] based on linear OA(27³¹, 732, F₂₇, 16) (dual of [732, 701, 17]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,13]), C₂Â =Â C([0,14]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,13]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,14]) [i] based on
    1. linear OA(27²⁹, 728, F₂₇, 15) (dual of [728, 699, 16]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,13}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(27²⁹, 728, F₂₇, 15) (dual of [728, 699, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(27³¹, 728, F₂₇, 16) (dual of [728, 697, 17]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,14}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    4. linear OA(27²⁷, 728, F₂₇, 14) (dual of [728, 701, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    5. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 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]
    6. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code) (see above)

(37−16, 37, 603630)-Net in Base 27 — Upper bound on s

There is no (21, 37, 603631)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 91297 830674 705960 478914 939765 870410 959492 033316 375089 > 27³⁷ [i]

Best Known (37−16, 37, s)-Nets in Base 27

(37−16, 37, 164)-Net over F27 — Constructive and digital

(37−16, 37, 216)-Net in Base 27 — Constructive

(37−16, 37, 885)-Net over F27 — Digital

(37−16, 37, 603630)-Net in Base 27 — Upper bound on s

(37−16, 37, 164)-Net over F₂₇ — Constructive and digital

(37−16, 37, 885)-Net over F₂₇ — Digital