MinT - Best Known (45, 55, s)-Nets in Base 3

Best Known (45, 55, s)-Nets in Base 3

Digital (45, 55, 3936)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁵⁵, 3936, F₃, 10, 10) (dual of [(3936, 10), 39305, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3⁵⁵, 19680, F₃, 10) (dual of [19680, 19625, 11]-code), using
  - discarding factors / shortening the dual code based on linear OA(3⁵⁵, 19683, F₃, 10) (dual of [19683, 19628, 11]-code), using
    - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

Digital (45, 55, 6918)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁵⁵, 6918, F₃, 2, 10) (dual of [(6918, 2), 13781, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3⁵⁵, 9841, F₃, 2, 10) (dual of [(9841, 2), 19627, 11]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3⁵⁵, 19682, F₃, 10) (dual of [19682, 19627, 11]-code), using
    - discarding factors / shortening the dual code based on linear OA(3⁵⁵, 19683, F₃, 10) (dual of [19683, 19628, 11]-code), using
      - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

There is no (45, 55, 230745)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 174 450643 251924 111486 909219 > 3⁵⁵ [i]