MinT - Best Known (47−11, 47, s)-Nets in Base 128

Best Known (47−11, 47, s)-Nets in Base 128

(47−11, 47, 1677870)-Net over F₁₂₈ — Constructive and digital

Digital (36, 47, 1677870)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 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 (30, 41, 1677720)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴¹, 1677720, F₁₂₈, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(128⁴¹, 8388601, F₁₂₈, 11) (dual of [8388601, 8388560, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴¹, large, F₁₂₈, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 15790321Â | 128⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(47−11, 47, 1726873)-Net in Base 128 — Constructive

(36, 47, 1726873)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (6, 11, 49153)-net in base 128, using
  - net defined by OOA [i] based on OOA(128¹¹, 49153, S₁₂₈, 5, 5), using
    - appending kth column [i] based on OOA(128¹¹, 49153, S₁₂₈, 4, 5), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(128³, 16513, F₁₂₈, 4, 2) (dual of [(16513, 4), 66049, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(128³, 16513, F₁₂₈, 2, 2) (dual of [(16513, 2), 33023, 3]-NRT-code), using
        appending kth column [i] based on linear OA(128³, 16513, F₁₂₈, 2) (dual of [16513, 16510, 3]-code), using
        Hamming code H(3,128) [i]
        OOA(128⁸, 32640, S₁₂₈, 4, 5), using
        OOA 2-folding and stacking with additional row [i] based on OA(128⁸, 65281, S₁₂₈, 5), using
        discarding parts of the base [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
2. (25, 36, 1677720)-net in base 128, using
  - net defined by OOA [i] based on OOA(128³⁶, 1677720, S₁₂₈, 11, 11), using
    - OOA 5-folding and stacking with additional row [i] based on OA(128³⁶, 8388601, S₁₂₈, 11), using
      - discarding factors based on OA(128³⁶, large, S₁₂₈, 11), using
        discarding parts of the base [i] based on linear OA(256³¹, large, F₂₅₆, 11) (dual of [large, large−31, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(47−11, 47, large)-Net over F₁₂₈ — Digital

Digital (36, 47, large)-net over F₁₂₈, using

t-expansion [i] based on digital (35, 47, large)-net over F₁₂₈, using
- 1 times m-reduction [i] based on digital (35, 48, large)-net over F₁₂₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(47−11, 47, large)-Net in Base 128 — Upper bound on s

There is no (36, 47, large)-net in base 128, because

9 times m-reduction [i] would yield (36, 38, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (47−11, 47, s)-Nets in Base 128

(47−11, 47, 1677870)-Net over F128 — Constructive and digital

(47−11, 47, 1726873)-Net in Base 128 — Constructive

(47−11, 47, large)-Net over F128 — Digital

(47−11, 47, large)-Net in Base 128 — Upper bound on s

(47−11, 47, 1677870)-Net over F₁₂₈ — Constructive and digital

(47−11, 47, large)-Net over F₁₂₈ — Digital