MinT - Best Known (119−30, 119, s)-Nets in Base 3

Best Known (119−30, 119, s)-Nets in Base 3

(119−30, 119, 264)-Net over F₃ — Constructive and digital

Digital (89, 119, 264)-net over F₃, using

1 times m-reduction [i] based on digital (89, 120, 264)-net over F₃, using
- trace code for nets [i] based on digital (9, 40, 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]

(119−30, 119, 555)-Net over F₃ — Digital

Digital (89, 119, 555)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁹, 555, F₃, 30) (dual of [555, 436, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁹, 739, F₃, 30) (dual of [739, 620, 31]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(27) [i] based on
    1. linear OA(3¹¹⁸, 729, F₃, 31) (dual of [729, 611, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(3¹⁰⁹, 729, F₃, 28) (dual of [729, 620, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(3¹, 10, F₃, 1) (dual of [10, 9, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(119−30, 119, 19569)-Net in Base 3 — Upper bound on s

There is no (89, 119, 19570)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 599 288788 692021 636769 855849 359122 815413 644223 847944 935385 > 3¹¹⁹ [i]