MinT - Best Known (54−9, 54, s)-Nets in Base 4

Best Known (54−9, 54, s)-Nets in Base 4

(54−9, 54, 16394)-Net over F₄ — Constructive and digital

Digital (45, 54, 16394)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 10)-net over F₄, using
  - linear code embedding found in a computer search [i]
2. digital (40, 49, 16384)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁴⁹, 16384, F₄, 9, 9) (dual of [(16384, 9), 147407, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(4⁴⁹, 65537, F₄, 9) (dual of [65537, 65488, 10]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(54−9, 54, 40757)-Net over F₄ — Digital

Digital (45, 54, 40757)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁴, 40757, F₄, 9) (dual of [40757, 40703, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵⁴, 65548, F₄, 9) (dual of [65548, 65494, 10]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4⁵, 11, F₄, 4) (dual of [11, 6, 5]-code), using
      - quaternary constacyclic codes with minimum distance 5 [i]
    2. linear OA(4⁴⁹, 65537, F₄, 9) (dual of [65537, 65488, 10]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(54−9, 54, large)-Net in Base 4 — Upper bound on s

There is no (45, 54, large)-net in base 4, because

7 times m-reduction [i] would yield (45, 47, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]