MinT - Best Known (60−22, 60, s)-Nets in Base 128

Best Known (60−22, 60, s)-Nets in Base 128

(60−22, 60, 1768)-Net over F₁₂₈ — Constructive and digital

Digital (38, 60, 1768)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (6, 17, 279)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 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, 12, 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 (21, 43, 1489)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴³, 1489, F₁₂₈, 22, 22) (dual of [(1489, 22), 32715, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(128⁴³, 16379, F₁₂₈, 22) (dual of [16379, 16336, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴³, 16384, F₁₂₈, 22) (dual of [16384, 16341, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(60−22, 60, 5960)-Net in Base 128 — Constructive

(38, 60, 5960)-net in base 128, using

1 times m-reduction [i] based on (38, 61, 5960)-net in base 128, using
- net defined by OOA [i] based on OOA(128⁶¹, 5960, S₁₂₈, 23, 23), using
  - OOA 11-folding and stacking with additional row [i] based on OA(128⁶¹, 65561, S₁₂₈, 23), using
    - discarding factors based on OA(128⁶¹, 65562, S₁₂₈, 23), using
      - discarding parts of the base [i] based on linear OA(256⁵³, 65562, F₂₅₆, 23) (dual of [65562, 65509, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(13) [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(256²⁷, 65536, F₂₅₆, 14) (dual of [65536, 65509, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(256⁸, 26, F₂₅₆, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,256)), using
        discarding factors / shortening the dual code based on linear OA(256⁸, 256, F₂₅₆, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
        Reedâ€“Solomon code RS(248,256) [i]

(60−22, 60, 71672)-Net over F₁₂₈ — Digital

Digital (38, 60, 71672)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(60−22, 60, large)-Net in Base 128 — Upper bound on s

There is no (38, 60, large)-net in base 128, because

20 times m-reduction [i] would yield (38, 40, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (60−22, 60, s)-Nets in Base 128

(60−22, 60, 1768)-Net over F128 — Constructive and digital

(60−22, 60, 5960)-Net in Base 128 — Constructive

(60−22, 60, 71672)-Net over F128 — Digital

(60−22, 60, large)-Net in Base 128 — Upper bound on s

(60−22, 60, 1768)-Net over F₁₂₈ — Constructive and digital

(60−22, 60, 71672)-Net over F₁₂₈ — Digital