MinT - Best Known (64−12, 64, s)-Nets in Base 128

Best Known (64−12, 64, s)-Nets in Base 128

(64−12, 64, 2097280)-Net over F₁₂₈ — Constructive and digital

Digital (52, 64, 2097280)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (13, 19, 699180)-net over F₁₂₈, using
  - (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 (10, 16, 699051)-net over F₁₂₈, using
      - net defined by OOA [i] based on linear OOA(128¹⁶, 699051, F₁₂₈, 6, 6) (dual of [(699051, 6), 4194290, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(128¹⁶, 2097153, F₁₂₈, 6) (dual of [2097153, 2097137, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹⁶, 2097155, F₁₂₈, 6) (dual of [2097155, 2097139, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(128¹⁶, 2097152, F₁₂₈, 6) (dual of [2097152, 2097136, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(128¹³, 2097152, F₁₂₈, 5) (dual of [2097152, 2097139, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (33, 45, 1398100)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴⁵, 1398100, F₁₂₈, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(128⁴⁵, 8388600, F₁₂₈, 12) (dual of [8388600, 8388555, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴⁵, large, F₁₂₈, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9256395Â | 128⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(64−12, 64, 2828840)-Net in Base 128 — Constructive

(52, 64, 2828840)-net in base 128, using

base change [i] based on digital (44, 56, 2828840)-net over F₂₅₆, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (2, 6, 32640)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁶, 32640, F₂₅₆, 4, 4) (dual of [(32640, 4), 130554, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (10, 16, 1398100)-net over F₂₅₆, using
    - s-reduction based on digital (10, 16, 2796201)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256¹⁶, 2796201, F₂₅₆, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(256¹⁶, large, F₂₅₆, 6) (dual of [large, large−16, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  3. digital (22, 34, 1398100)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256³⁴, 1398100, F₂₅₆, 12, 12) (dual of [(1398100, 12), 16777166, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(256³⁴, 8388600, F₂₅₆, 12) (dual of [8388600, 8388566, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(64−12, 64, large)-Net over F₁₂₈ — Digital

Digital (52, 64, large)-net over F₁₂₈, using

t-expansion [i] based on digital (51, 64, large)-net over F₁₂₈, using
- 6 times m-reduction [i] based on digital (51, 70, large)-net over F₁₂₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(64−12, 64, large)-Net in Base 128 — Upper bound on s

There is no (52, 64, large)-net in base 128, because

10 times m-reduction [i] would yield (52, 54, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (64−12, 64, s)-Nets in Base 128

(64−12, 64, 2097280)-Net over F128 — Constructive and digital

(64−12, 64, 2828840)-Net in Base 128 — Constructive

(64−12, 64, large)-Net over F128 — Digital

(64−12, 64, large)-Net in Base 128 — Upper bound on s

(64−12, 64, 2097280)-Net over F₁₂₈ — Constructive and digital

(64−12, 64, large)-Net over F₁₂₈ — Digital