MinT - Best Known (31, 49, s)-Nets in Base 128

Best Known (31, 49, s)-Nets in Base 128

(31, 49, 2099)-Net over F₁₂₈ — Constructive and digital

Digital (31, 49, 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 (17, 35, 1820)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128³⁵, 1820, F₁₂₈, 18, 18) (dual of [(1820, 18), 32725, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(128³⁵, 16380, F₁₂₈, 18) (dual of [16380, 16345, 19]-code), using
      - discarding factors / shortening the dual code based on linear OA(128³⁵, 16384, F₁₂₈, 18) (dual of [16384, 16349, 19]-code), using
        an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(31, 49, 7411)-Net in Base 128 — Constructive

(31, 49, 7411)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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. (22, 40, 7282)-net in base 128, using
  - base change [i] based on digital (17, 35, 7282)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256³⁵, 7282, F₂₅₆, 18, 18) (dual of [(7282, 18), 131041, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(256³⁵, 65538, F₂₅₆, 18) (dual of [65538, 65503, 19]-code), using
        constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
        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(256³³, 65536, F₂₅₆, 17) (dual of [65536, 65503, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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]

(31, 49, 66976)-Net over F₁₂₈ — Digital

Digital (31, 49, 66976)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 49, large)-Net in Base 128 — Upper bound on s

There is no (31, 49, large)-net in base 128, because

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

Best Known (31, 49, s)-Nets in Base 128

(31, 49, 2099)-Net over F128 — Constructive and digital

(31, 49, 7411)-Net in Base 128 — Constructive

(31, 49, 66976)-Net over F128 — Digital

(31, 49, large)-Net in Base 128 — Upper bound on s

(31, 49, 2099)-Net over F₁₂₈ — Constructive and digital

(31, 49, 66976)-Net over F₁₂₈ — Digital