MinT - Best Known (51−19, 51, s)-Nets in Base 128

Best Known (51−19, 51, s)-Nets in Base 128

(51−19, 51, 2099)-Net over F₁₂₈ — Constructive and digital

Digital (32, 51, 2099)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 279)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
    2. digital (1, 10, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
2. digital (18, 37, 1820)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128³⁷, 1820, F₁₂₈, 19, 19) (dual of [(1820, 19), 34543, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(128³⁷, 16381, F₁₂₈, 19) (dual of [16381, 16344, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(128³⁷, 16384, F₁₂₈, 19) (dual of [16384, 16347, 20]-code), using
        an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(51−19, 51, 7284)-Net in Base 128 — Constructive

(32, 51, 7284)-net in base 128, using

net defined by OOA [i] based on OOA(128⁵¹, 7284, S₁₂₈, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(128⁵¹, 65557, S₁₂₈, 19), using
  - discarding factors based on OA(128⁵¹, 65560, S₁₂₈, 19), using
    - discarding parts of the base [i] based on linear OA(256⁴⁴, 65560, F₂₅₆, 19) (dual of [65560, 65516, 20]-code), using
      - constructionÂ X applied to C([0,9]) âŠ‚ C([0,5]) [i] based on
        linear OA(256³⁷, 65537, F₂₅₆, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(256²¹, 65537, F₂₅₆, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(256⁷, 23, F₂₅₆, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,256)), using
        discarding factors / shortening the dual code based on linear OA(256⁷, 256, F₂₅₆, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
        Reedâ€“Solomon code RS(249,256) [i]

(51−19, 51, 55568)-Net over F₁₂₈ — Digital

Digital (32, 51, 55568)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−19, 51, large)-Net in Base 128 — Upper bound on s

There is no (32, 51, large)-net in base 128, because

17 times m-reduction [i] would yield (32, 34, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (51−19, 51, s)-Nets in Base 128

(51−19, 51, 2099)-Net over F128 — Constructive and digital

(51−19, 51, 7284)-Net in Base 128 — Constructive

(51−19, 51, 55568)-Net over F128 — Digital

(51−19, 51, large)-Net in Base 128 — Upper bound on s

(51−19, 51, 2099)-Net over F₁₂₈ — Constructive and digital

(51−19, 51, 55568)-Net over F₁₂₈ — Digital