MinT - Best Known (74, 74+24, s)-Nets in Base 3

Best Known (74, 74+24, s)-Nets in Base 3

(74, 74+24, 328)-Net over F₃ — Constructive and digital

Digital (74, 98, 328)-net over F₃, using

3² times duplication [i] based on digital (72, 96, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 24, 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]

(74, 74+24, 556)-Net over F₃ — Digital

Digital (74, 98, 556)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁸, 556, F₃, 24) (dual of [556, 458, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁸, 742, F₃, 24) (dual of [742, 644, 25]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
    1. linear OA(3⁹⁷, 729, F₃, 25) (dual of [729, 632, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(3⁸⁵, 729, F₃, 22) (dual of [729, 644, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(3¹, 13, F₃, 1) (dual of [13, 12, 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]

(74, 74+24, 20824)-Net in Base 3 — Upper bound on s

There is no (74, 98, 20825)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 57268 995125 201816 626639 372223 000592 703644 219441 > 3⁹⁸ [i]