MinT - Best Known (25, 39, s)-Nets in Base 128

Best Known (25, 39, s)-Nets in Base 128

(25, 39, 2727)-Net over F₁₂₈ — Constructive and digital

Digital (25, 39, 2727)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 387)-net over F₁₂₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 129)-net over F₁₂₈, using
      - kÂ =Â 2 construction [i]
    2. 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]
    3. digital (0, 7, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. digital (13, 27, 2340)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128²⁷, 2340, F₁₂₈, 14, 14) (dual of [(2340, 14), 32733, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(128²⁷, 16380, F₁₂₈, 14) (dual of [16380, 16353, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(128²⁷, 16384, F₁₂₈, 14) (dual of [16384, 16357, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(25, 39, 9619)-Net in Base 128 — Constructive

(25, 39, 9619)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (1, 8, 257)-net in base 128, using
  - base change [i] based on digital (0, 7, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
2. (17, 31, 9362)-net in base 128, using
  - net defined by OOA [i] based on OOA(128³¹, 9362, S₁₂₈, 14, 14), using
    - OA 7-folding and stacking [i] based on OA(128³¹, 65534, S₁₂₈, 14), using
      - discarding factors based on OA(128³¹, 65538, S₁₂₈, 14), using
        discarding parts of the base [i] based on linear OA(256²⁷, 65538, F₂₅₆, 14) (dual of [65538, 65511, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(256²⁷, 65536, F₂₅₆, 14) (dual of [65536, 65509, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(256²⁵, 65536, F₂₅₆, 13) (dual of [65536, 65511, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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]

(25, 39, 93597)-Net over F₁₂₈ — Digital

Digital (25, 39, 93597)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 39, large)-Net in Base 128 — Upper bound on s

There is no (25, 39, large)-net in base 128, because

12 times m-reduction [i] would yield (25, 27, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (25, 39, s)-Nets in Base 128

(25, 39, 2727)-Net over F128 — Constructive and digital

(25, 39, 9619)-Net in Base 128 — Constructive

(25, 39, 93597)-Net over F128 — Digital

(25, 39, large)-Net in Base 128 — Upper bound on s

(25, 39, 2727)-Net over F₁₂₈ — Constructive and digital

(25, 39, 93597)-Net over F₁₂₈ — Digital