MinT - Best Known (89, 89+16, s)-Nets in Base 4

Best Known (89, 89+16, s)-Nets in Base 4

(89, 89+16, 8201)-Net over F₄ — Constructive and digital

Digital (89, 105, 8201)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (80, 96, 8192)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁹⁶, 8192, F₄, 16, 16) (dual of [(8192, 16), 130976, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(4⁹⁶, 65536, F₄, 16) (dual of [65536, 65440, 17]-code), using
      - 1 times truncation [i] based on linear OA(4⁹⁷, 65537, F₄, 17) (dual of [65537, 65440, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(89, 89+16, 59800)-Net over F₄ — Digital

Digital (89, 105, 59800)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁵, 59800, F₄, 16) (dual of [59800, 59695, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁰⁵, 65577, F₄, 16) (dual of [65577, 65472, 17]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹⁰⁴, 65576, F₄, 16) (dual of [65576, 65472, 17]-code), using
    - constructionÂ X applied to C([0,8]) âŠ‚ C([0,5]) [i] based on
      1. linear OA(4⁹⁷, 65537, F₄, 17) (dual of [65537, 65440, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      2. linear OA(4⁶⁵, 65537, F₄, 11) (dual of [65537, 65472, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
      3. linear OA(4⁷, 39, F₄, 4) (dual of [39, 32, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(89, 89+16, large)-Net in Base 4 — Upper bound on s

There is no (89, 105, large)-net in base 4, because

14 times m-reduction [i] would yield (89, 91, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]