MinT - Best Known (128−32, 128, s)-Nets in Base 3

Best Known (128−32, 128, s)-Nets in Base 3

(128−32, 128, 328)-Net over F₃ — Constructive and digital

Digital (96, 128, 328)-net over F₃, using

trace code for nets [i] based on digital (0, 32, 82)-net over F₈₁, using
- net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
    - the rational function field F₈₁(x) [i]
  - Niederreiter sequence [i]

(128−32, 128, 604)-Net over F₃ — Digital

Digital (96, 128, 604)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁸, 604, F₃, 32) (dual of [604, 476, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁸, 745, F₃, 32) (dual of [745, 617, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(28) [i] based on
    1. linear OA(3¹²⁴, 729, F₃, 32) (dual of [729, 605, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    2. linear OA(3¹¹², 729, F₃, 29) (dual of [729, 617, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(3⁴, 16, F₃, 2) (dual of [16, 12, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(128−32, 128, 22293)-Net in Base 3 — Upper bound on s

There is no (96, 128, 22294)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 794955 981214 144916 285077 540063 156236 763887 728083 070561 014945 > 3¹²⁸ [i]