MinT - Best Known (131−16, 131, s)-Nets in Base 4

Best Known (131−16, 131, s)-Nets in Base 4

(131−16, 131, 131086)-Net over F₄ — Constructive and digital

Digital (115, 131, 131086)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (104, 120, 131072)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹²⁰, 131072, F₄, 16, 16) (dual of [(131072, 16), 2097032, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(4¹²⁰, 1048576, F₄, 16) (dual of [1048576, 1048456, 17]-code), using
      - 1 times truncation [i] based on linear OA(4¹²¹, 1048577, F₄, 17) (dual of [1048577, 1048456, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(131−16, 131, 785033)-Net over F₄ — Digital

Digital (115, 131, 785033)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³¹, 785033, F₄, 16) (dual of [785033, 784902, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³¹, 1048636, F₄, 16) (dual of [1048636, 1048505, 17]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(9) [i] based on
    1. linear OA(4¹²¹, 1048576, F₄, 17) (dual of [1048576, 1048455, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(4⁷¹, 1048576, F₄, 10) (dual of [1048576, 1048505, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(4¹⁰, 60, F₄, 5) (dual of [60, 50, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(131−16, 131, large)-Net in Base 4 — Upper bound on s

There is no (115, 131, large)-net in base 4, because

14 times m-reduction [i] would yield (115, 117, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]