MinT - Best Known (124−23, 124, s)-Nets in Base 3

Best Known (124−23, 124, s)-Nets in Base 3

(124−23, 124, 688)-Net over F₃ — Constructive and digital

Digital (101, 124, 688)-net over F₃, using

t-expansion [i] based on digital (100, 124, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 31, 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]

(124−23, 124, 3287)-Net over F₃ — Digital

Digital (101, 124, 3287)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹²⁴, 3287, F₃, 2, 23) (dual of [(3287, 2), 6450, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹²⁴, 6574, F₃, 23) (dual of [6574, 6450, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(19) [i] based on
    1. linear OA(3¹²¹, 6561, F₃, 23) (dual of [6561, 6440, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(3¹⁰⁵, 6561, F₃, 20) (dual of [6561, 6456, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(124−23, 124, 530956)-Net in Base 3 — Upper bound on s

There is no (101, 124, 530957)-net in base 3, because

1 times m-reduction [i] would yield (101, 123, 530957)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 48520 086249 291078 798194 069806 454884 178047 205019 558247 550955 > 3¹²³ [i]