MinT - Best Known (48−5, 48, s)-Nets in Base 3

Best Known (48−5, 48, s)-Nets in Base 3

(48−5, 48, 4194305)-Net over F₃ — Constructive and digital

Digital (43, 48, 4194305)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁴⁸, 4194305, F₃, 5, 5) (dual of [(4194305, 5), 20971477, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁴⁸, 4194305, F₃, 4, 5) (dual of [(4194305, 4), 16777172, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3², 4, F₃, 4, 2) (dual of [(4, 4), 14, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(4;14,3) [i]
    2. linear OOA(3⁴⁶, 4194301, F₃, 4, 5) (dual of [(4194301, 4), 16777158, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(3⁴⁶, large, F₃, 5) (dual of [large, large−46, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(48−5, 48, large)-Net over F₃ — Digital

Digital (43, 48, large)-net over F₃, using

3² times duplication [i] based on digital (41, 46, large)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3⁴⁶, large, F₃, 5, 5), using
  - appending kth column [i] based on linear OOA(3⁴⁶, large, F₃, 4, 5), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁴⁶, large, F₃, 5) (dual of [large, large−46, 6]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(48−5, 48, large)-Net in Base 3 — Upper bound on s

There is no (43, 48, large)-net in base 3, because

3 times m-reduction [i] would yield (43, 45, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]