MinT - Best Known (78−28, 78, s)-Nets in Base 128

Best Known (78−28, 78, s)-Nets in Base 128

(78−28, 78, 1470)-Net over F₁₂₈ — Constructive and digital

Digital (50, 78, 1470)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 23, 300)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 8, 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, 15, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)
2. digital (27, 55, 1170)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁵⁵, 1170, F₁₂₈, 28, 28) (dual of [(1170, 28), 32705, 29]-NRT-code), using
    - OA 14-folding and stacking [i] based on linear OA(128⁵⁵, 16380, F₁₂₈, 28) (dual of [16380, 16325, 29]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁵⁵, 16384, F₁₂₈, 28) (dual of [16384, 16329, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(78−28, 78, 4831)-Net in Base 128 — Constructive

(50, 78, 4831)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 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. (35, 63, 4681)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁶³, 4681, S₁₂₈, 28, 28), using
    - OA 14-folding and stacking [i] based on OA(128⁶³, 65534, S₁₂₈, 28), using
      - discarding factors based on OA(128⁶³, 65538, S₁₂₈, 28), using
        discarding parts of the base [i] based on linear OA(256⁵⁵, 65538, F₂₅₆, 28) (dual of [65538, 65483, 29]-code), using
        constructionÂ X applied to C_e(27) âŠ‚ C_e(26) [i] based on
        linear OA(256⁵⁵, 65536, F₂₅₆, 28) (dual of [65536, 65481, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(256⁵³, 65536, F₂₅₆, 27) (dual of [65536, 65483, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [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]

(78−28, 78, 105234)-Net over F₁₂₈ — Digital

Digital (50, 78, 105234)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−28, 78, large)-Net in Base 128 — Upper bound on s

There is no (50, 78, large)-net in base 128, because

26 times m-reduction [i] would yield (50, 52, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (78−28, 78, s)-Nets in Base 128

(78−28, 78, 1470)-Net over F128 — Constructive and digital

(78−28, 78, 4831)-Net in Base 128 — Constructive

(78−28, 78, 105234)-Net over F128 — Digital

(78−28, 78, large)-Net in Base 128 — Upper bound on s

(78−28, 78, 1470)-Net over F₁₂₈ — Constructive and digital

(78−28, 78, 105234)-Net over F₁₂₈ — Digital