MinT - Best Known (135−12, 135, s)-Nets in Base 4

Best Known (135−12, 135, s)-Nets in Base 4

Digital (123, 135, 4194301)-net over F₄, using

net defined by OOA [i] based on linear OOA(4¹³⁵, 4194301, F₄, 15, 12) (dual of [(4194301, 15), 62914380, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(4¹³⁵, large, F₄, 3, 12), using
  - trace code [i] based on linear OOA(64⁴⁵, 2796201, F₆₄, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

Digital (123, 135, large)-net over F₄, using

t-expansion [i] based on digital (121, 135, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³⁵, large, F₄, 14) (dual of [large, large−135, 15]-code), using
  - 14 times code embedding in larger space [i] based on linear OA(4¹²¹, large, F₄, 14) (dual of [large, large−121, 15]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

There is no (123, 135, large)-net in base 4, because

10 times m-reduction [i] would yield (123, 125, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]