MinT - Best Known (55, 55+19, s)-Nets in Base 3

Best Known (55, 55+19, s)-Nets in Base 3

(55, 55+19, 204)-Net over F₃ — Constructive and digital

Digital (55, 74, 204)-net over F₃, using

1 times m-reduction [i] based on digital (55, 75, 204)-net over F₃, using
- trace code for nets [i] based on digital (5, 25, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(55, 55+19, 387)-Net over F₃ — Digital

Digital (55, 74, 387)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁴, 387, F₃, 19) (dual of [387, 313, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁴, 736, F₃, 19) (dual of [736, 662, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(16) [i] based on
    1. linear OA(3⁷³, 729, F₃, 19) (dual of [729, 656, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(3⁶⁷, 729, F₃, 17) (dual of [729, 662, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(3¹, 7, F₃, 1) (dual of [7, 6, 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]

(55, 55+19, 15362)-Net in Base 3 — Upper bound on s

There is no (55, 74, 15363)-net in base 3, because

1 times m-reduction [i] would yield (55, 73, 15363)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 67600 930072 758074 136167 235700 338103 > 3⁷³ [i]