MinT - Best Known (76−16, 76, s)-Nets in Base 3

Best Known (76−16, 76, s)-Nets in Base 3

(76−16, 76, 464)-Net over F₃ — Constructive and digital

Digital (60, 76, 464)-net over F₃, using

t-expansion [i] based on digital (59, 76, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 19, 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]

(76−16, 76, 1103)-Net over F₃ — Digital

Digital (60, 76, 1103)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁷⁶, 1103, F₃, 2, 16) (dual of [(1103, 2), 2130, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁷⁶, 2206, F₃, 16) (dual of [2206, 2130, 17]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3⁷⁵, 2205, F₃, 16) (dual of [2205, 2130, 17]-code), using
    - constructionÂ X applied to C_e(15) âŠ‚ C_e(12) [i] based on
      1. linear OA(3⁷¹, 2187, F₃, 16) (dual of [2187, 2116, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      2. linear OA(3⁵⁷, 2187, F₃, 13) (dual of [2187, 2130, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      3. linear OA(3⁴, 18, F₃, 2) (dual of [18, 14, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(76−16, 76, 64159)-Net in Base 3 — Upper bound on s

There is no (60, 76, 64160)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 824905 483469 154079 347030 855136 712705 > 3⁷⁶ [i]