MinT - Best Known (67−13, 67, s)-Nets in Base 16

Best Known (67−13, 67, s)-Nets in Base 16

(67−13, 67, 174780)-Net over F₁₆ — Constructive and digital

Digital (54, 67, 174780)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (48, 61, 174763)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁶¹, 174763, F₁₆, 13, 13) (dual of [(174763, 13), 2271858, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(16⁶¹, 1048579, F₁₆, 13) (dual of [1048579, 1048518, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁶¹, 1048581, F₁₆, 13) (dual of [1048581, 1048520, 14]-code), using
        constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(16⁶¹, 1048576, F₁₆, 13) (dual of [1048576, 1048515, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(16⁵⁶, 1048576, F₁₆, 12) (dual of [1048576, 1048520, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(16⁰, 5, F₁₆, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(67−13, 67, 349526)-Net in Base 16 — Constructive

(54, 67, 349526)-net in base 16, using

net defined by OOA [i] based on OOA(16⁶⁷, 349526, S₁₆, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(16⁶⁷, 2097157, S₁₆, 13), using
  - discarding factors based on OA(16⁶⁷, 2097160, S₁₆, 13), using
    - discarding parts of the base [i] based on linear OA(128³⁸, 2097160, F₁₂₈, 13) (dual of [2097160, 2097122, 14]-code), using
      - constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(128³⁷, 2097153, F₁₂₈, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(128³¹, 2097153, F₁₂₈, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(128¹, 7, F₁₂₈, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(67−13, 67, 1863266)-Net over F₁₆ — Digital

Digital (54, 67, 1863266)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−13, 67, large)-Net in Base 16 — Upper bound on s

There is no (54, 67, large)-net in base 16, because

11 times m-reduction [i] would yield (54, 56, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (67−13, 67, s)-Nets in Base 16

(67−13, 67, 174780)-Net over F16 — Constructive and digital

(67−13, 67, 349526)-Net in Base 16 — Constructive

(67−13, 67, 1863266)-Net over F16 — Digital

(67−13, 67, large)-Net in Base 16 — Upper bound on s

(67−13, 67, 174780)-Net over F₁₆ — Constructive and digital

(67−13, 67, 1863266)-Net over F₁₆ — Digital