MinT - Best Known (51, 51+12, s)-Nets in Base 4

Best Known (51, 51+12, s)-Nets in Base 4

Digital (51, 63, 2730)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁶³, 2730, F₄, 12, 12) (dual of [(2730, 12), 32697, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4⁶³, 16380, F₄, 12) (dual of [16380, 16317, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁶³, 16383, F₄, 12) (dual of [16383, 16320, 13]-code), using
    - 1 times truncation [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]

Digital (51, 63, 8191)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4⁶³, 8191, F₄, 2, 12) (dual of [(8191, 2), 16319, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4⁶³, 16382, F₄, 12) (dual of [16382, 16319, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁶³, 16383, F₄, 12) (dual of [16383, 16320, 13]-code), using
    - 1 times truncation [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, 63, 2092811)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 85 070834 080808 801744 779351 189913 449090 > 4⁶³ [i]