MinT - Best Known (127, 127+29, s)-Nets in Base 3

Best Known (127, 127+29, s)-Nets in Base 3

(127, 127+29, 688)-Net over F₃ — Constructive and digital

Digital (127, 156, 688)-net over F₃, using

4 times m-reduction [i] based on digital (127, 160, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 40, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(127, 127+29, 3287)-Net over F₃ — Digital

Digital (127, 156, 3287)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵⁶, 3287, F₃, 2, 29) (dual of [(3287, 2), 6418, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁵⁶, 6574, F₃, 29) (dual of [6574, 6418, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(25) [i] based on
    1. linear OA(3¹⁵³, 6561, F₃, 29) (dual of [6561, 6408, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(3¹³⁷, 6561, F₃, 26) (dual of [6561, 6424, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(127, 127+29, 579203)-Net in Base 3 — Upper bound on s

There is no (127, 156, 579204)-net in base 3, because

1 times m-reduction [i] would yield (127, 155, 579204)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 89 907316 041705 253736 046297 516149 843550 072294 543527 661476 355667 034044 408537 > 3¹⁵⁵ [i]