MinT - Best Known (24, 24+8, s)-Nets in Base 3

Best Known (24, 24+8, s)-Nets in Base 3

(24, 24+8, 328)-Net over F₃ — Constructive and digital

Digital (24, 32, 328)-net over F₃, using

trace code for nets [i] based on digital (0, 8, 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]

(24, 24+8, 432)-Net over F₃ — Digital

Digital (24, 32, 432)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³², 432, F₃, 8) (dual of [432, 400, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(3³², 737, F₃, 8) (dual of [737, 705, 9]-code), using
  - constructionÂ X4 applied to C([0,7]) âŠ‚ C([1,6]) [i] based on
    1. linear OA(3³¹, 728, F₃, 8) (dual of [728, 697, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    2. linear OA(3²⁴, 728, F₃, 6) (dual of [728, 704, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    3. linear OA(3⁸, 9, F₃, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
      - dual of repetition code with length 9 [i]
    4. linear OA(3¹, 9, F₃, 1) (dual of [9, 8, 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]

(24, 24+8, 7257)-Net in Base 3 — Upper bound on s

There is no (24, 32, 7258)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1853 591114 459049 > 3³² [i]