MinT - Best Known (76−20, 76, s)-Nets in Base 128

Best Known (76−20, 76, s)-Nets in Base 128

(76−20, 76, 210102)-Net over F₁₂₈ — Constructive and digital

Digital (56, 76, 210102)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 387)-net over F₁₂₈, using
  - 1 times m-reduction [i] based on 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 (38, 58, 209715)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁵⁸, 209715, F₁₂₈, 20, 20) (dual of [(209715, 20), 4194242, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(128⁵⁸, 2097150, F₁₂₈, 20) (dual of [2097150, 2097092, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁵⁸, 2097152, F₁₂₈, 20) (dual of [2097152, 2097094, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(76−20, 76, 838860)-Net in Base 128 — Constructive

(56, 76, 838860)-net in base 128, using

128⁴ times duplication [i] based on (52, 72, 838860)-net in base 128, using
- t-expansion [i] based on (51, 72, 838860)-net in base 128, using
  - base change [i] based on digital (42, 63, 838860)-net over F₂₅₆, using
    - 256² times duplication [i] based on digital (40, 61, 838860)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁶¹, 838860, F₂₅₆, 21, 21) (dual of [(838860, 21), 17615999, 22]-NRT-code), using
        OOA 10-folding and stacking with additional row [i] based on linear OA(256⁶¹, 8388601, F₂₅₆, 21) (dual of [8388601, 8388540, 22]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁶¹, large, F₂₅₆, 21) (dual of [large, large−61, 22]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(76−20, 76, large)-Net over F₁₂₈ — Digital

Digital (56, 76, large)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (56, 77, large)-net over F₁₂₈, using
- Gilbertâ€“VarÅ¡amov bound [i]

(76−20, 76, large)-Net in Base 128 — Upper bound on s

There is no (56, 76, large)-net in base 128, because

18 times m-reduction [i] would yield (56, 58, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (76−20, 76, s)-Nets in Base 128

(76−20, 76, 210102)-Net over F128 — Constructive and digital

(76−20, 76, 838860)-Net in Base 128 — Constructive

(76−20, 76, large)-Net over F128 — Digital

(76−20, 76, large)-Net in Base 128 — Upper bound on s

(76−20, 76, 210102)-Net over F₁₂₈ — Constructive and digital

(76−20, 76, large)-Net over F₁₂₈ — Digital