MinT - Best Known (55, 75, s)-Nets in Base 3

Best Known (55, 75, s)-Nets in Base 3

(55, 75, 204)-Net over F₃ — Constructive and digital

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, 75, 314)-Net over F₃ — Digital

Digital (55, 75, 314)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁵, 314, F₃, 20) (dual of [314, 239, 21]-code), using
- 57 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 14 times 0) [i] based on linear OA(3⁶⁶, 248, F₃, 20) (dual of [248, 182, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
    1. linear OA(3⁶⁶, 243, F₃, 20) (dual of [243, 177, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(3⁶¹, 243, F₃, 19) (dual of [243, 182, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(3⁰, 5, F₃, 0) (dual of [5, 5, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(55, 75, 8567)-Net in Base 3 — Upper bound on s

There is no (55, 75, 8568)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 608333 813083 923464 098428 067743 494097 > 3⁷⁵ [i]