MinT - Best Known (51, 51+13, s)-Nets in Base 4

Best Known (51, 51+13, s)-Nets in Base 4

Digital (51, 64, 2730)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁶⁴, 2730, F₄, 13, 13) (dual of [(2730, 13), 35426, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4⁶⁴, 16381, F₄, 13) (dual of [16381, 16317, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁶⁴, 16384, F₄, 13) (dual of [16384, 16320, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

Digital (51, 64, 8151)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4⁶⁴, 8151, F₄, 2, 13) (dual of [(8151, 2), 16238, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4⁶⁴, 8192, F₄, 2, 13) (dual of [(8192, 2), 16320, 14]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(4⁶⁴, 16384, F₄, 13) (dual of [16384, 16320, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

There is no (51, 64, 2092811)-net in base 4, because

1 times m-reduction [i] would yield (51, 63, 2092811)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 85 070834 080808 801744 779351 189913 449090 > 4⁶³ [i]