MinT - Best Known (116−28, 116, s)-Nets in Base 3

Best Known (116−28, 116, s)-Nets in Base 3

(116−28, 116, 400)-Net over F₃ — Constructive and digital

Digital (88, 116, 400)-net over F₃, using

trace code for nets [i] based on digital (1, 29, 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]

(116−28, 116, 657)-Net over F₃ — Digital

Digital (88, 116, 657)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁶, 657, F₃, 28) (dual of [657, 541, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁶, 754, F₃, 28) (dual of [754, 638, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(22) [i] based on
    1. linear OA(3¹⁰⁹, 729, F₃, 28) (dual of [729, 620, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [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(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]

(116−28, 116, 27133)-Net in Base 3 — Upper bound on s

There is no (88, 116, 27134)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 193650 495190 423844 045596 045135 230981 595977 336171 552053 > 3¹¹⁶ [i]