MinT - Best Known (106−21, 106, s)-Nets in Base 4

Best Known (106−21, 106, s)-Nets in Base 4

Digital (85, 106, 1638)-net over F₄, using

net defined by OOA [i] based on linear OOA(4¹⁰⁶, 1638, F₄, 21, 21) (dual of [(1638, 21), 34292, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4¹⁰⁶, 16381, F₄, 21) (dual of [16381, 16275, 22]-code), using
  - discarding factors / shortening the dual code based on linear OA(4¹⁰⁶, 16384, F₄, 21) (dual of [16384, 16278, 22]-code), using
    - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

Digital (85, 106, 7564)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁰⁶, 7564, F₄, 2, 21) (dual of [(7564, 2), 15022, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4¹⁰⁶, 8192, F₄, 2, 21) (dual of [(8192, 2), 16278, 22]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(4¹⁰⁶, 16384, F₄, 21) (dual of [16384, 16278, 22]-code), using
    - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

There is no (85, 106, 3165803)-net in base 4, because

1 times m-reduction [i] would yield (85, 105, 3165803)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1645 504685 940356 566771 557947 856311 473265 792936 248968 776001 317341 > 4¹⁰⁵ [i]