MinT - Best Known (42−19, 42, s)-Nets in Base 27

Best Known (42−19, 42, s)-Nets in Base 27

(42−19, 42, 164)-Net over F₂₇ — Constructive and digital

Digital (23, 42, 164)-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 (7, 26, 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]

(42−19, 42, 216)-Net in Base 27 — Constructive

(23, 42, 216)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (5, 14, 100)-net in base 27, using
  - 2 times m-reduction [i] based on (5, 16, 100)-net in base 27, using
    - base change [i] based on digital (1, 12, 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, 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]

(42−19, 42, 767)-Net over F₂₇ — Digital

Digital (23, 42, 767)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴², 767, F₂₇, 19) (dual of [767, 725, 20]-code), using
- 30 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 6 times 0, 1, 20 times 0) [i] based on linear OA(27³⁷, 732, F₂₇, 19) (dual of [732, 695, 20]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,16]), C₂Â =Â C([0,17]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,16]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,17]) [i] based on
    1. linear OA(27³⁵, 728, F₂₇, 18) (dual of [728, 693, 19]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(27³⁵, 728, F₂₇, 18) (dual of [728, 693, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(27³⁷, 728, F₂₇, 19) (dual of [728, 691, 20]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,17}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    4. linear OA(27³³, 728, F₂₇, 17) (dual of [728, 695, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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)

(42−19, 42, 528970)-Net in Base 27 — Upper bound on s

There is no (23, 42, 528971)-net in base 27, because

1 times m-reduction [i] would yield (23, 41, 528971)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 48519 462566 084154 101945 797825 440674 971649 168231 067795 579791 > 27⁴¹ [i]

Best Known (42−19, 42, s)-Nets in Base 27

(42−19, 42, 164)-Net over F27 — Constructive and digital

(42−19, 42, 216)-Net in Base 27 — Constructive

(42−19, 42, 767)-Net over F27 — Digital

(42−19, 42, 528970)-Net in Base 27 — Upper bound on s

(42−19, 42, 164)-Net over F₂₇ — Constructive and digital

(42−19, 42, 767)-Net over F₂₇ — Digital