MinT - Best Known (53−19, 53, s)-Nets in Base 128

Best Known (53−19, 53, s)-Nets in Base 128

(53−19, 53, 2207)-Net over F₁₂₈ — Constructive and digital

Digital (34, 53, 2207)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 16, 387)-net over F₁₂₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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, 4, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
    3. 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]

(53−19, 53, 7431)-Net in Base 128 — Constructive

(34, 53, 7431)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. 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. (24, 43, 7281)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁴³, 7281, S₁₂₈, 19, 19), using
    - OOA 9-folding and stacking with additional row [i] based on OA(128⁴³, 65530, S₁₂₈, 19), using
      - discarding factors based on OA(128⁴³, 65538, S₁₂₈, 19), using
        discarding parts of the base [i] based on linear OA(256³⁷, 65538, F₂₅₆, 19) (dual of [65538, 65501, 20]-code), using
        constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
        linear OA(256³⁷, 65536, F₂₅₆, 19) (dual of [65536, 65499, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(256³⁵, 65536, F₂₅₆, 18) (dual of [65536, 65501, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(53−19, 53, 95264)-Net over F₁₂₈ — Digital

Digital (34, 53, 95264)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(53−19, 53, large)-Net in Base 128 — Upper bound on s

There is no (34, 53, large)-net in base 128, because

17 times m-reduction [i] would yield (34, 36, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (53−19, 53, s)-Nets in Base 128

(53−19, 53, 2207)-Net over F128 — Constructive and digital

(53−19, 53, 7431)-Net in Base 128 — Constructive

(53−19, 53, 95264)-Net over F128 — Digital

(53−19, 53, large)-Net in Base 128 — Upper bound on s

(53−19, 53, 2207)-Net over F₁₂₈ — Constructive and digital

(53−19, 53, 95264)-Net over F₁₂₈ — Digital