MinT - Best Known (124−29, 124, s)-Nets in Base 3

Best Known (124−29, 124, s)-Nets in Base 3

(124−29, 124, 464)-Net over F₃ — Constructive and digital

Digital (95, 124, 464)-net over F₃, using

trace code for nets [i] based on digital (2, 31, 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]

(124−29, 124, 788)-Net over F₃ — Digital

Digital (95, 124, 788)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁴, 788, F₃, 29) (dual of [788, 664, 30]-code), using
- 44 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0) [i] based on linear OA(3¹¹², 732, F₃, 29) (dual of [732, 620, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
    1. linear OA(3¹¹², 729, F₃, 29) (dual of [729, 617, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. 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]
    3. linear OA(3⁰, 3, F₃, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(124−29, 124, 47006)-Net in Base 3 — Upper bound on s

There is no (95, 124, 47007)-net in base 3, because

1 times m-reduction [i] would yield (95, 123, 47007)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 48532 512891 245622 194441 617812 541929 706418 266808 175289 121317 > 3¹²³ [i]