MinT - Best Known (45−19, 45, s)-Nets in Base 27

Best Known (45−19, 45, s)-Nets in Base 27

(45−19, 45, 184)-Net over F₂₇ — Constructive and digital

Digital (26, 45, 184)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 9, 56)-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 (0, 6, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (4, 13, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
3. digital (4, 23, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(45−19, 45, 250)-Net in Base 27 — Constructive

(26, 45, 250)-net in base 27, using

27¹ times duplication [i] based on (25, 44, 250)-net in base 27, using
- base change [i] based on digital (14, 33, 250)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 10, 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 (4, 23, 150)-net over F₈₁, using
      - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(45−19, 45, 1111)-Net over F₂₇ — Digital

Digital (26, 45, 1111)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴⁵, 1111, F₂₇, 19) (dual of [1111, 1066, 20]-code), using
- 371 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 6 times 0, 1, 20 times 0, 1, 54 times 0, 1, 112 times 0, 1, 172 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)

(45−19, 45, 1586921)-Net in Base 27 — Upper bound on s

There is no (26, 45, 1586922)-net in base 27, because

1 times m-reduction [i] would yield (26, 44, 1586922)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 955 010248 336754 816564 932327 535919 578286 236436 748964 747240 081653 > 27⁴⁴ [i]

Best Known (45−19, 45, s)-Nets in Base 27

(45−19, 45, 184)-Net over F27 — Constructive and digital

(45−19, 45, 250)-Net in Base 27 — Constructive

(45−19, 45, 1111)-Net over F27 — Digital

(45−19, 45, 1586921)-Net in Base 27 — Upper bound on s

(45−19, 45, 184)-Net over F₂₇ — Constructive and digital

(45−19, 45, 1111)-Net over F₂₇ — Digital