MinT - Best Known (80, 80+26, s)-Nets in Base 3

Best Known (80, 80+26, s)-Nets in Base 3

(80, 80+26, 328)-Net over F₃ — Constructive and digital

Digital (80, 106, 328)-net over F₃, using

3² times duplication [i] based on digital (78, 104, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 26, 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]

(80, 80+26, 578)-Net over F₃ — Digital

Digital (80, 106, 578)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁶, 578, F₃, 26) (dual of [578, 472, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰⁶, 742, F₃, 26) (dual of [742, 636, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(22) [i] based on
    1. linear OA(3¹⁰³, 729, F₃, 26) (dual of [729, 626, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    2. 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]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(80, 80+26, 22004)-Net in Base 3 — Upper bound on s

There is no (80, 106, 22005)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 375 917834 143919 827668 395935 554135 779158 783224 325339 > 3¹⁰⁶ [i]