MinT - Best Known (97−23, 97, s)-Nets in Base 3

Best Known (97−23, 97, s)-Nets in Base 3

(97−23, 97, 400)-Net over F₃ — Constructive and digital

Digital (74, 97, 400)-net over F₃, using

3¹ times duplication [i] based on digital (73, 96, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 24, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(97−23, 97, 640)-Net over F₃ — Digital

Digital (74, 97, 640)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁷, 640, F₃, 23) (dual of [640, 543, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁷, 753, F₃, 23) (dual of [753, 656, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(18) [i] based on
    1. linear OA(3⁹¹, 729, F₃, 23) (dual of [729, 638, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(3⁷³, 729, F₃, 19) (dual of [729, 656, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(3⁶, 24, F₃, 3) (dual of [24, 18, 4]-code or 24-cap in PG(5,3)), using
      - discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(97−23, 97, 35795)-Net in Base 3 — Upper bound on s

There is no (74, 97, 35796)-net in base 3, because

1 times m-reduction [i] would yield (74, 96, 35796)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6364 358748 923904 226114 004271 697208 672839 966513 > 3⁹⁶ [i]