MinT - Best Known (86, 86+29, s)-Nets in Base 3

Best Known (86, 86+29, s)-Nets in Base 3

(86, 86+29, 264)-Net over F₃ — Constructive and digital

Digital (86, 115, 264)-net over F₃, using

3¹ times duplication [i] based on digital (85, 114, 264)-net over F₃, using
- trace code for nets [i] based on digital (9, 38, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(86, 86+29, 542)-Net over F₃ — Digital

Digital (86, 115, 542)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁵, 542, F₃, 29) (dual of [542, 427, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁵, 728, F₃, 29) (dual of [728, 613, 30]-code), using
  - the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {−6,−5,…,22}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(86, 86+29, 23190)-Net in Base 3 — Upper bound on s

There is no (86, 115, 23191)-net in base 3, because

1 times m-reduction [i] would yield (86, 114, 23191)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 466162 747986 650157 632200 322376 506503 191825 716363 953237 > 3¹¹⁴ [i]