MinT - Best Known (61, 80, s)-Nets in Base 3

Best Known (61, 80, s)-Nets in Base 3

(61, 80, 400)-Net over F₃ — Constructive and digital

Digital (61, 80, 400)-net over F₃, using

trace code for nets [i] based on digital (1, 20, 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]

(61, 80, 577)-Net over F₃ — Digital

Digital (61, 80, 577)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸⁰, 577, F₃, 19) (dual of [577, 497, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁸⁰, 754, F₃, 19) (dual of [754, 674, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(13) [i] based on
    1. 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]
    2. linear OA(3⁵⁵, 729, F₃, 14) (dual of [729, 674, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(3⁷, 25, F₃, 4) (dual of [25, 18, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁷, 26, F₃, 4) (dual of [26, 19, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(61, 80, 31964)-Net in Base 3 — Upper bound on s

There is no (61, 80, 31965)-net in base 3, because

1 times m-reduction [i] would yield (61, 79, 31965)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 49 274068 730862 218989 134500 224082 442907 > 3⁷⁹ [i]