MinT - Best Known (104−11, 104, s)-Nets in Base 4

Best Known (104−11, 104, s)-Nets in Base 4

(104−11, 104, 1677736)-Net over F₄ — Constructive and digital

Digital (93, 104, 1677736)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 16)-net over F₄, using
  - linear code embedding found in a computer search [i]
2. digital (86, 97, 1677720)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁹⁷, 1677720, F₄, 11, 11) (dual of [(1677720, 11), 18454823, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(4⁹⁷, 8388601, F₄, 11) (dual of [8388601, 8388504, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁹⁷, large, F₄, 11) (dual of [large, large−97, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(104−11, 104, large)-Net over F₄ — Digital

Digital (93, 104, large)-net over F₄, using

4¹ times duplication [i] based on digital (92, 103, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰³, large, F₄, 11) (dual of [large, large−103, 12]-code), using
  - 6 times code embedding in larger space [i] based on linear OA(4⁹⁷, large, F₄, 11) (dual of [large, large−97, 12]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(104−11, 104, large)-Net in Base 4 — Upper bound on s

There is no (93, 104, large)-net in base 4, because

9 times m-reduction [i] would yield (93, 95, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]