MinT - Best Known (88, 118, s)-Nets in Base 3

Best Known (88, 118, s)-Nets in Base 3

(88, 118, 264)-Net over F₃ — Constructive and digital

Digital (88, 118, 264)-net over F₃, using

3¹ times duplication [i] based on digital (87, 117, 264)-net over F₃, using
- trace code for nets [i] based on digital (9, 39, 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]

(88, 118, 533)-Net over F₃ — Digital

Digital (88, 118, 533)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁸, 533, F₃, 30) (dual of [533, 415, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁸, 728, F₃, 30) (dual of [728, 610, 31]-code), using
  - the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {336,337,…,365}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(88, 118, 18186)-Net in Base 3 — Upper bound on s

There is no (88, 118, 18187)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 199 781970 658804 820903 936295 824831 498901 405359 810912 833811 > 3¹¹⁸ [i]