MinT - Best Known (64, 81, s)-Nets in Base 3

Best Known (64, 81, s)-Nets in Base 3

(64, 81, 464)-Net over F₃ — Constructive and digital

Digital (64, 81, 464)-net over F₃, using

3¹ times duplication [i] based on digital (63, 80, 464)-net over F₃, using
- t-expansion [i] based on digital (62, 80, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 20, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(64, 81, 1112)-Net over F₃ — Digital

Digital (64, 81, 1112)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸¹, 1112, F₃, 17) (dual of [1112, 1031, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁸¹, 2200, F₃, 17) (dual of [2200, 2119, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [i] based on
    1. linear OA(3⁷⁸, 2187, F₃, 17) (dual of [2187, 2109, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(3⁶⁴, 2187, F₃, 14) (dual of [2187, 2123, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(64, 81, 111133)-Net in Base 3 — Upper bound on s

There is no (64, 81, 111134)-net in base 3, because

1 times m-reduction [i] would yield (64, 80, 111134)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 147 818729 812110 003828 233192 668974 027345 > 3⁸⁰ [i]