MinT - Best Known (57, 57+6, s)-Nets in Base 3

Best Known (57, 57+6, s)-Nets in Base 3

(57, 57+6, 2796205)-Net over F₃ — Constructive and digital

Digital (57, 63, 2796205)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁶³, 2796205, F₃, 6, 6) (dual of [(2796205, 6), 16777167, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁶³, 2796205, F₃, 5, 6) (dual of [(2796205, 5), 13980962, 7]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3³, 4, F₃, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(5;17,3) [i]
    2. linear OOA(3⁶⁰, 2796201, F₃, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(57, 57+6, large)-Net over F₃ — Digital

Digital (57, 63, large)-net over F₃, using

3³ times duplication [i] based on digital (54, 60, large)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3⁶⁰, large, F₃, 6, 6), using
  - appending kth column [i] based on linear OOA(3⁶⁰, large, F₃, 5, 6), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(57, 57+6, large)-Net in Base 3 — Upper bound on s

There is no (57, 63, large)-net in base 3, because

4 times m-reduction [i] would yield (57, 59, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]