MinT - Best Known (111−19, 111, s)-Nets in Base 4

Best Known (111−19, 111, s)-Nets in Base 4

(111−19, 111, 1835)-Net over F₄ — Constructive and digital

Digital (92, 111, 1835)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 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 (80, 99, 1821)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁹⁹, 1821, F₄, 19, 19) (dual of [(1821, 19), 34500, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(4⁹⁹, 16390, F₄, 19) (dual of [16390, 16291, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁹⁹, 16391, F₄, 19) (dual of [16391, 16292, 20]-code), using
        constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
        linear OA(4⁹⁹, 16384, F₄, 19) (dual of [16384, 16285, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(4⁹², 16384, F₄, 18) (dual of [16384, 16292, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(4⁰, 7, F₄, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(111−19, 111, 16433)-Net over F₄ — Digital

Digital (92, 111, 16433)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹¹, 16433, F₄, 19) (dual of [16433, 16322, 20]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(4¹⁰⁹, 16429, F₄, 19) (dual of [16429, 16320, 20]-code), using
    - constructionÂ X applied to C_e(18) âŠ‚ C_e(12) [i] based on
      1. linear OA(4⁹⁹, 16384, F₄, 19) (dual of [16384, 16285, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]
      3. linear OA(4¹⁰, 45, F₄, 5) (dual of [45, 35, 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]
  2. linear OA(4¹⁰⁹, 16431, F₄, 18) (dual of [16431, 16322, 19]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 4¹⁰⁹Â > V_b^sâˆ’1(kâˆ’1)Â = 166872 685441 752513 722190 940821 059033 293816 304441 635352 717448 085054 [i]
  3. linear OA(4⁰, 2, F₄, 0) (dual of [2, 2, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(111−19, 111, large)-Net in Base 4 — Upper bound on s

There is no (92, 111, large)-net in base 4, because

17 times m-reduction [i] would yield (92, 94, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (111−19, 111, s)-Nets in Base 4

(111−19, 111, 1835)-Net over F4 — Constructive and digital

(111−19, 111, 16433)-Net over F4 — Digital

(111−19, 111, large)-Net in Base 4 — Upper bound on s

(111−19, 111, 1835)-Net over F₄ — Constructive and digital

(111−19, 111, 16433)-Net over F₄ — Digital