MinT - Best Known (24, 24+21, s)-Nets in Base 27

Best Known (24, 24+21, s)-Nets in Base 27

(24, 24+21, 164)-Net over F₂₇ — Constructive and digital

Digital (24, 45, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 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]
2. digital (7, 28, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(24, 24+21, 200)-Net in Base 27 — Constructive

(24, 45, 200)-net in base 27, using

27¹ times duplication [i] based on (23, 44, 200)-net in base 27, using
- base change [i] based on digital (12, 33, 200)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 11, 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 (1, 22, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)

(24, 24+21, 621)-Net over F₂₇ — Digital

Digital (24, 45, 621)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴⁵, 621, F₂₇, 21) (dual of [621, 576, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁴⁵, 743, F₂₇, 21) (dual of [743, 698, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
    1. linear OA(27⁴¹, 729, F₂₇, 21) (dual of [729, 688, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(27³¹, 729, F₂₇, 16) (dual of [729, 698, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(27⁴, 14, F₂₇, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,27)), using
      - discarding factors / shortening the dual code based on linear OA(27⁴, 27, F₂₇, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
        Reedâ€“Solomon code RS(23,27) [i]

(24, 24+21, 345937)-Net in Base 27 — Upper bound on s

There is no (24, 45, 345938)-net in base 27, because

1 times m-reduction [i] would yield (24, 44, 345938)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 955 024407 246032 368960 637777 434347 186821 585833 775695 688746 082053 > 27⁴⁴ [i]