MinT - Best Known (76, 76+25, s)-Nets in Base 3

Best Known (76, 76+25, s)-Nets in Base 3

(76, 76+25, 328)-Net over F₃ — Constructive and digital

Digital (76, 101, 328)-net over F₃, using

3¹ times duplication [i] based on digital (75, 100, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 25, 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]

(76, 76+25, 540)-Net over F₃ — Digital

Digital (76, 101, 540)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰¹, 540, F₃, 25) (dual of [540, 439, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰¹, 740, F₃, 25) (dual of [740, 639, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(3⁹⁷, 730, F₃, 25) (dual of [730, 633, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. linear OA(3⁸⁵, 730, F₃, 21) (dual of [730, 645, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    3. linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
      - ovoid in PG(3,Â 3) [i]

(76, 76+25, 25011)-Net in Base 3 — Upper bound on s

There is no (76, 101, 25012)-net in base 3, because

1 times m-reduction [i] would yield (76, 100, 25012)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 515464 982369 161886 765802 371755 609046 011639 803777 > 3¹⁰⁰ [i]