MinT - Best Known (61−22, 61, s)-Nets in Base 128

Best Known (61−22, 61, s)-Nets in Base 128

(61−22, 61, 1789)-Net over F₁₂₈ — Constructive and digital

Digital (39, 61, 1789)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 18, 300)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 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 (1, 12, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)
2. digital (21, 43, 1489)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴³, 1489, F₁₂₈, 22, 22) (dual of [(1489, 22), 32715, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(128⁴³, 16379, F₁₂₈, 22) (dual of [16379, 16336, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴³, 16384, F₁₂₈, 22) (dual of [16384, 16341, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(61−22, 61, 6087)-Net in Base 128 — Constructive

(39, 61, 6087)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 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. (28, 50, 5958)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁵⁰, 5958, S₁₂₈, 22, 22), using
    - OA 11-folding and stacking [i] based on OA(128⁵⁰, 65538, S₁₂₈, 22), using
      - discarding parts of the base [i] based on linear OA(256⁴³, 65538, F₂₅₆, 22) (dual of [65538, 65495, 23]-code), using
        constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
        linear OA(256⁴³, 65536, F₂₅₆, 22) (dual of [65536, 65493, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(256⁴¹, 65536, F₂₅₆, 21) (dual of [65536, 65495, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(61−22, 61, 90299)-Net over F₁₂₈ — Digital

Digital (39, 61, 90299)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(61−22, 61, large)-Net in Base 128 — Upper bound on s

There is no (39, 61, large)-net in base 128, because

20 times m-reduction [i] would yield (39, 41, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (61−22, 61, s)-Nets in Base 128

(61−22, 61, 1789)-Net over F128 — Constructive and digital

(61−22, 61, 6087)-Net in Base 128 — Constructive

(61−22, 61, 90299)-Net over F128 — Digital

(61−22, 61, large)-Net in Base 128 — Upper bound on s

(61−22, 61, 1789)-Net over F₁₂₈ — Constructive and digital

(61−22, 61, 90299)-Net over F₁₂₈ — Digital