MinT - Best Known (46−24, 46, s)-Nets in Base 27

Best Known (46−24, 46, s)-Nets in Base 27

(46−24, 46, 140)-Net over F₂₇ — Constructive and digital

Digital (22, 46, 140)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 16, 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]
2. digital (6, 30, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(46−24, 46, 172)-Net in Base 27 — Constructive

(22, 46, 172)-net in base 27, using

14 times m-reduction [i] based on (22, 60, 172)-net in base 27, using
- base change [i] based on digital (7, 45, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(46−24, 46, 286)-Net over F₂₇ — Digital

Digital (22, 46, 286)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴⁶, 286, F₂₇, 24) (dual of [286, 240, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁴⁶, 364, F₂₇, 24) (dual of [364, 318, 25]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 364Â | 27²âˆ’1, defining interval IÂ =Â [0,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(46−24, 46, 62408)-Net in Base 27 — Upper bound on s

There is no (22, 46, 62409)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 696318 103963 149730 908673 111429 254622 791793 097193 605578 720266 340497 > 27⁴⁶ [i]