MinT - Best Known (63−23, 63, s)-Nets in Base 128

Best Known (63−23, 63, s)-Nets in Base 128

(63−23, 63, 1789)-Net over F₁₂₈ — Constructive and digital

Digital (40, 63, 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 (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]

(63−23, 63, 6086)-Net in Base 128 — Constructive

(40, 63, 6086)-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. (29, 52, 5957)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁵², 5957, S₁₂₈, 23, 23), using
    - OOA 11-folding and stacking with additional row [i] based on OA(128⁵², 65528, S₁₂₈, 23), using
      - discarding factors based on OA(128⁵², 65538, S₁₂₈, 23), using
        discarding parts of the base [i] based on linear OA(256⁴⁵, 65538, F₂₅₆, 23) (dual of [65538, 65493, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] 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]
        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(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]

(63−23, 63, 77160)-Net over F₁₂₈ — Digital

Digital (40, 63, 77160)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(63−23, 63, large)-Net in Base 128 — Upper bound on s

There is no (40, 63, large)-net in base 128, because

21 times m-reduction [i] would yield (40, 42, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (63−23, 63, s)-Nets in Base 128

(63−23, 63, 1789)-Net over F128 — Constructive and digital

(63−23, 63, 6086)-Net in Base 128 — Constructive

(63−23, 63, 77160)-Net over F128 — Digital

(63−23, 63, large)-Net in Base 128 — Upper bound on s

(63−23, 63, 1789)-Net over F₁₂₈ — Constructive and digital

(63−23, 63, 77160)-Net over F₁₂₈ — Digital