MinT - Best Known (64−23, 64, s)-Nets in Base 128

Best Known (64−23, 64, s)-Nets in Base 128

(64−23, 64, 1876)-Net over F₁₂₈ — Constructive and digital

Digital (41, 64, 1876)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 19, 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, 5, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
    3. digital (0, 11, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. digital (22, 45, 1489)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴⁵, 1489, F₁₂₈, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(128⁴⁵, 16380, F₁₂₈, 23) (dual of [16380, 16335, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴⁵, 16384, F₁₂₈, 23) (dual of [16384, 16339, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(64−23, 64, 6214)-Net in Base 128 — Constructive

(41, 64, 6214)-net in base 128, using

base change [i] based on digital (33, 56, 6214)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 11, 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. digital (22, 45, 5957)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁵, 5957, F₂₅₆, 23, 23) (dual of [(5957, 23), 136966, 24]-NRT-code), using
      - OOA 11-folding and stacking with additional row [i] based on linear OA(256⁴⁵, 65528, F₂₅₆, 23) (dual of [65528, 65483, 24]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁵, 65536, F₂₅₆, 23) (dual of [65536, 65491, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(64−23, 64, 96198)-Net over F₁₂₈ — Digital

Digital (41, 64, 96198)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−23, 64, large)-Net in Base 128 — Upper bound on s

There is no (41, 64, large)-net in base 128, because

21 times m-reduction [i] would yield (41, 43, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (64−23, 64, s)-Nets in Base 128

(64−23, 64, 1876)-Net over F128 — Constructive and digital

(64−23, 64, 6214)-Net in Base 128 — Constructive

(64−23, 64, 96198)-Net over F128 — Digital

(64−23, 64, large)-Net in Base 128 — Upper bound on s

(64−23, 64, 1876)-Net over F₁₂₈ — Constructive and digital

(64−23, 64, 96198)-Net over F₁₂₈ — Digital