MinT - Best Known (90, 90+24, s)-Nets in Base 3

Best Known (90, 90+24, s)-Nets in Base 3

(90, 90+24, 600)-Net over F₃ — Constructive and digital

Digital (90, 114, 600)-net over F₃, using

3² times duplication [i] based on digital (88, 112, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 28, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(90, 90+24, 1258)-Net over F₃ — Digital

Digital (90, 114, 1258)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁴, 1258, F₃, 24) (dual of [1258, 1144, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁴, 2202, F₃, 24) (dual of [2202, 2088, 25]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
    1. linear OA(3¹¹³, 2187, F₃, 25) (dual of [2187, 2074, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(3⁹⁹, 2187, F₃, 22) (dual of [2187, 2088, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 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]

(90, 90+24, 90142)-Net in Base 3 — Upper bound on s

There is no (90, 114, 90143)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 465311 818316 075766 516345 268208 239627 979019 251437 609449 > 3¹¹⁴ [i]