MinT - Best Known (50−19, 50, s)-Nets in Base 128

Best Known (50−19, 50, s)-Nets in Base 128

(50−19, 50, 2078)-Net over F₁₂₈ — Constructive and digital

Digital (31, 50, 2078)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 258)-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 (0, 9, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
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]

(50−19, 50, 7283)-Net in Base 128 — Constructive

(31, 50, 7283)-net in base 128, using

128² times duplication [i] based on (29, 48, 7283)-net in base 128, using
- base change [i] based on digital (23, 42, 7283)-net over F₂₅₆, using
  - 256² times duplication [i] based on digital (21, 40, 7283)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 7283, F₂₅₆, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
      - OOA 9-folding and stacking with additional row [i] based on linear OA(256⁴⁰, 65548, F₂₅₆, 19) (dual of [65548, 65508, 20]-code), using
        constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [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(256²⁹, 65537, F₂₅₆, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(50−19, 50, 42440)-Net over F₁₂₈ — Digital

Digital (31, 50, 42440)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−19, 50, large)-Net in Base 128 — Upper bound on s

There is no (31, 50, large)-net in base 128, because

17 times m-reduction [i] would yield (31, 33, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (50−19, 50, s)-Nets in Base 128

(50−19, 50, 2078)-Net over F128 — Constructive and digital

(50−19, 50, 7283)-Net in Base 128 — Constructive

(50−19, 50, 42440)-Net over F128 — Digital

(50−19, 50, large)-Net in Base 128 — Upper bound on s

(50−19, 50, 2078)-Net over F₁₂₈ — Constructive and digital

(50−19, 50, 42440)-Net over F₁₂₈ — Digital