MinT - Best Known (58−21, 58, s)-Nets in Base 128

Best Known (58−21, 58, s)-Nets in Base 128

(58−21, 58, 1938)-Net over F₁₂₈ — Constructive and digital

Digital (37, 58, 1938)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 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, 11, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)
2. digital (20, 41, 1638)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴¹, 1638, F₁₂₈, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(128⁴¹, 16381, F₁₂₈, 21) (dual of [16381, 16340, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴¹, 16384, F₁₂₈, 21) (dual of [16384, 16343, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(58−21, 58, 6703)-Net in Base 128 — Constructive

(37, 58, 6703)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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. (26, 47, 6553)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁴⁷, 6553, S₁₂₈, 21, 21), using
    - OOA 10-folding and stacking with additional row [i] based on OA(128⁴⁷, 65531, S₁₂₈, 21), using
      - discarding factors based on OA(128⁴⁷, 65538, S₁₂₈, 21), using
        discarding parts of the base [i] based on linear OA(256⁴¹, 65538, F₂₅₆, 21) (dual of [65538, 65497, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        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(256³⁹, 65536, F₂₅₆, 20) (dual of [65536, 65497, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [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]

(58−21, 58, 84423)-Net over F₁₂₈ — Digital

Digital (37, 58, 84423)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(58−21, 58, large)-Net in Base 128 — Upper bound on s

There is no (37, 58, large)-net in base 128, because

19 times m-reduction [i] would yield (37, 39, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (58−21, 58, s)-Nets in Base 128

(58−21, 58, 1938)-Net over F128 — Constructive and digital

(58−21, 58, 6703)-Net in Base 128 — Constructive

(58−21, 58, 84423)-Net over F128 — Digital

(58−21, 58, large)-Net in Base 128 — Upper bound on s

(58−21, 58, 1938)-Net over F₁₂₈ — Constructive and digital

(58−21, 58, 84423)-Net over F₁₂₈ — Digital