MinT - Best Known (153−29, 153, s)-Nets in Base 3

Best Known (153−29, 153, s)-Nets in Base 3

(153−29, 153, 688)-Net over F₃ — Constructive and digital

Digital (124, 153, 688)-net over F₃, using

3 times m-reduction [i] based on digital (124, 156, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 39, 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]

(153−29, 153, 3089)-Net over F₃ — Digital

Digital (124, 153, 3089)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵³, 3089, F₃, 2, 29) (dual of [(3089, 2), 6025, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁵³, 3284, F₃, 2, 29) (dual of [(3284, 2), 6415, 30]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁵³, 6568, F₃, 29) (dual of [6568, 6415, 30]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹⁵³, 6569, F₃, 29) (dual of [6569, 6416, 30]-code), using
      - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
        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]
        linear OA(3¹⁴⁵, 6561, F₃, 28) (dual of [6561, 6416, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(3⁰, 8, F₃, 0) (dual of [8, 8, 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]

(153−29, 153, 457708)-Net in Base 3 — Upper bound on s

There is no (124, 153, 457709)-net in base 3, because

1 times m-reduction [i] would yield (124, 152, 457709)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 329992 408372 651693 710598 875017 899430 967478 018747 500356 696237 377484 514033 > 3¹⁵² [i]