MinT - Best Known (138−14, 138, s)-Nets in Base 3

Best Known (138−14, 138, s)-Nets in Base 3

Digital (124, 138, 1198371)-net over F₃, using

3² times duplication [i] based on digital (122, 136, 1198371)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹³⁶, 1198371, F₃, 14, 14) (dual of [(1198371, 14), 16777058, 15]-NRT-code), using
  - OA 7-folding and stacking [i] based on linear OA(3¹³⁶, 8388597, F₃, 14) (dual of [8388597, 8388461, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

Digital (124, 138, 2796201)-net over F₃, using

3² times duplication [i] based on digital (122, 136, 2796201)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³⁶, 2796201, F₃, 3, 14) (dual of [(2796201, 3), 8388467, 15]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

There is no (124, 138, large)-net in base 3, because

12 times m-reduction [i] would yield (124, 126, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]