MinT - Best Known (71−26, 71, s)-Nets in Base 128

Best Known (71−26, 71, s)-Nets in Base 128

(71−26, 71, 1539)-Net over F₁₂₈ — Constructive and digital

Digital (45, 71, 1539)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 20, 279)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 6, 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 (1, 14, 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 (25, 51, 1260)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁵¹, 1260, F₁₂₈, 26, 26) (dual of [(1260, 26), 32709, 27]-NRT-code), using
    - OA 13-folding and stacking [i] based on linear OA(128⁵¹, 16380, F₁₂₈, 26) (dual of [16380, 16329, 27]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁵¹, 16384, F₁₂₈, 26) (dual of [16384, 16333, 27]-code), using
        an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(71−26, 71, 5043)-Net in Base 128 — Constructive

(45, 71, 5043)-net in base 128, using

1 times m-reduction [i] based on (45, 72, 5043)-net in base 128, using
- base change [i] based on digital (36, 63, 5043)-net over F₂₅₆, using
  - 256³ times duplication [i] based on digital (33, 60, 5043)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁶⁰, 5043, F₂₅₆, 27, 27) (dual of [(5043, 27), 136101, 28]-NRT-code), using
      - OOA 13-folding and stacking with additional row [i] based on linear OA(256⁶⁰, 65560, F₂₅₆, 27) (dual of [65560, 65500, 28]-code), using
        constructionÂ X applied to C([0,13]) âŠ‚ C([0,9]) [i] based on
        linear OA(256⁵³, 65537, F₂₅₆, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(256³⁷, 65537, F₂₅₆, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(256⁷, 23, F₂₅₆, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,256)), using
        discarding factors / shortening the dual code based on linear OA(256⁷, 256, F₂₅₆, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
        Reedâ€“Solomon code RS(249,256) [i]

(71−26, 71, 77334)-Net over F₁₂₈ — Digital

Digital (45, 71, 77334)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−26, 71, large)-Net in Base 128 — Upper bound on s

There is no (45, 71, large)-net in base 128, because

24 times m-reduction [i] would yield (45, 47, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (71−26, 71, s)-Nets in Base 128

(71−26, 71, 1539)-Net over F128 — Constructive and digital

(71−26, 71, 5043)-Net in Base 128 — Constructive

(71−26, 71, 77334)-Net over F128 — Digital

(71−26, 71, large)-Net in Base 128 — Upper bound on s

(71−26, 71, 1539)-Net over F₁₂₈ — Constructive and digital

(71−26, 71, 77334)-Net over F₁₂₈ — Digital