MinT - Best Known (222−37, 222, s)-Nets in Base 3

Best Known (222−37, 222, s)-Nets in Base 3

(222−37, 222, 1480)-Net over F₃ — Constructive and digital

Digital (185, 222, 1480)-net over F₃, using

t-expansion [i] based on digital (184, 222, 1480)-net over F₃, using
- 2 times m-reduction [i] based on digital (184, 224, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 56, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(222−37, 222, 8241)-Net over F₃ — Digital

Digital (185, 222, 8241)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²², 8241, F₃, 2, 37) (dual of [(8241, 2), 16260, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²²², 9853, F₃, 2, 37) (dual of [(9853, 2), 19484, 38]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²²², 19706, F₃, 37) (dual of [19706, 19484, 38]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(3²²¹, 19705, F₃, 37) (dual of [19705, 19484, 38]-code), using
      - constructionÂ X applied to C_e(36) âŠ‚ C_e(33) [i] based on
        linear OA(3²¹⁷, 19683, F₃, 37) (dual of [19683, 19466, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
        linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
        linear OA(3⁴, 22, F₃, 2) (dual of [22, 18, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(222−37, 222, 2723240)-Net in Base 3 — Upper bound on s

There is no (185, 222, 2723241)-net in base 3, because

1 times m-reduction [i] would yield (185, 221, 2723241)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2778 424007 421171 649546 101207 735934 099440 622431 899776 222135 390197 111151 734751 383117 544542 636144 046259 753305 > 3²²¹ [i]