MinT - Best Known (72−13, 72, s)-Nets in Base 4

Best Known (72−13, 72, s)-Nets in Base 4

(72−13, 72, 2742)-Net over F₄ — Constructive and digital

Digital (59, 72, 2742)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 8, 12)-net over F₄, using
  - linear code embedding found in a computer search [i]
2. digital (51, 64, 2730)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁶⁴, 2730, F₄, 13, 13) (dual of [(2730, 13), 35426, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(4⁶⁴, 16381, F₄, 13) (dual of [16381, 16317, 14]-code), using
      - discarding factors / shortening the dual code 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]

(72−13, 72, 12579)-Net over F₄ — Digital

Digital (59, 72, 12579)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷², 12579, F₄, 13) (dual of [12579, 12507, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷², 16401, F₄, 13) (dual of [16401, 16329, 14]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4⁸, 17, F₄, 6) (dual of [17, 9, 7]-code), using
      - the BCH-code C(I) with length 17Â | 4⁴âˆ’1, defining interval IÂ =Â {1,4,7,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    2. 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]

(72−13, 72, large)-Net in Base 4 — Upper bound on s

There is no (59, 72, large)-net in base 4, because

11 times m-reduction [i] would yield (59, 61, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]