MinT - Best Known (42−18, 42, s)-Nets in Base 27

Best Known (42−18, 42, s)-Nets in Base 27

(42−18, 42, 168)-Net over F₂₇ — Constructive and digital

Digital (24, 42, 168)-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 (8, 26, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(42−18, 42, 232)-Net in Base 27 — Constructive

(24, 42, 232)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (6, 15, 116)-net in base 27, using
  - 1 times m-reduction [i] based on (6, 16, 116)-net in base 27, using
    - base change [i] based on digital (2, 12, 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. (9, 27, 116)-net in base 27, using
  - 1 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 (see above)

(42−18, 42, 968)-Net over F₂₇ — Digital

Digital (24, 42, 968)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴², 968, F₂₇, 18) (dual of [968, 926, 19]-code), using
- 229 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 7 times 0, 1, 24 times 0, 1, 63 times 0, 1, 128 times 0) [i] based on linear OA(27³⁵, 732, F₂₇, 18) (dual of [732, 697, 19]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,15]), C₂Â =Â C([0,16]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,15]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,16]) [i] based on
    1. linear OA(27³³, 728, F₂₇, 17) (dual of [728, 695, 18]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,15}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. 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]
    3. 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]
    4. linear OA(27³¹, 728, F₂₇, 16) (dual of [728, 697, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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)

(42−18, 42, 762909)-Net in Base 27 — Upper bound on s

There is no (24, 42, 762910)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 310023 641999 120046 730976 356207 730758 548230 849183 164389 818493 > 27⁴² [i]

Best Known (42−18, 42, s)-Nets in Base 27

(42−18, 42, 168)-Net over F27 — Constructive and digital

(42−18, 42, 232)-Net in Base 27 — Constructive

(42−18, 42, 968)-Net over F27 — Digital

(42−18, 42, 762909)-Net in Base 27 — Upper bound on s

(42−18, 42, 168)-Net over F₂₇ — Constructive and digital

(42−18, 42, 968)-Net over F₂₇ — Digital