MinT - Best Known (41, 65, s)-Nets in Base 128

Best Known (41, 65, s)-Nets in Base 128

(41, 65, 1623)-Net over F₁₂₈ — Constructive and digital

Digital (41, 65, 1623)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 258)-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 (0, 12, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. digital (23, 47, 1365)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴⁷, 1365, F₁₂₈, 24, 24) (dual of [(1365, 24), 32713, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(128⁴⁷, 16380, F₁₂₈, 24) (dual of [16380, 16333, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴⁷, 16384, F₁₂₈, 24) (dual of [16384, 16337, 25]-code), using
        an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(41, 65, 5463)-Net in Base 128 — Constructive

(41, 65, 5463)-net in base 128, using

128¹ times duplication [i] based on (40, 64, 5463)-net in base 128, using
- t-expansion [i] based on (39, 64, 5463)-net in base 128, using
  - base change [i] based on digital (31, 56, 5463)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁵⁶, 5463, F₂₅₆, 25, 25) (dual of [(5463, 25), 136519, 26]-NRT-code), using
      - OOA 12-folding and stacking with additional row [i] based on linear OA(256⁵⁶, 65557, F₂₅₆, 25) (dual of [65557, 65501, 26]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁵⁶, 65560, F₂₅₆, 25) (dual of [65560, 65504, 26]-code), using
        constructionÂ X applied to C([0,12]) âŠ‚ C([0,8]) [i] based on
        linear OA(256⁴⁹, 65537, F₂₅₆, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
        linear OA(256³³, 65537, F₂₅₆, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (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]

(41, 65, 66971)-Net over F₁₂₈ — Digital

Digital (41, 65, 66971)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 65, large)-Net in Base 128 — Upper bound on s

There is no (41, 65, large)-net in base 128, because

22 times m-reduction [i] would yield (41, 43, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (41, 65, s)-Nets in Base 128

(41, 65, 1623)-Net over F128 — Constructive and digital

(41, 65, 5463)-Net in Base 128 — Constructive

(41, 65, 66971)-Net over F128 — Digital

(41, 65, large)-Net in Base 128 — Upper bound on s

(41, 65, 1623)-Net over F₁₂₈ — Constructive and digital

(41, 65, 66971)-Net over F₁₂₈ — Digital