MinT - Best Known (69−35, 69, s)-Nets in Base 27

Best Known (69−35, 69, s)-Nets in Base 27

(69−35, 69, 176)-Net over F₂₇ — Constructive and digital

Digital (34, 69, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 24, 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 (10, 45, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(69−35, 69, 370)-Net in Base 27 — Constructive

(34, 69, 370)-net in base 27, using

3 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
- base change [i] based on digital (16, 54, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(69−35, 69, 436)-Net over F₂₇ — Digital

Digital (34, 69, 436)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁶⁹, 436, F₂₇, 35) (dual of [436, 367, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁶⁹, 728, F₂₇, 35) (dual of [728, 659, 36]-code), using
  - the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {3,4,…,37}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]

(69−35, 69, 146690)-Net in Base 27 — Upper bound on s

There is no (34, 69, 146691)-net in base 27, because

1 times m-reduction [i] would yield (34, 68, 146691)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 516563 458465 510015 264888 037270 353604 537089 445886 078865 837412 089623 384513 284984 480320 475869 449007 > 27⁶⁸ [i]