MinT - Best Known (37, 48, s)-Nets in Base 128

Best Known (37, 48, s)-Nets in Base 128

(37, 48, 1685848)-Net over F₁₂₈ — Constructive and digital

Digital (37, 48, 1685848)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 8128)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁷, 8128, F₁₂₈, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(128⁷, 8128, F₁₂₈, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(128⁷, 16257, F₁₂₈, 5) (dual of [16257, 16250, 6]-code), using
        a code by Danev and Olsson [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]

(37, 48, 1775641)-Net in Base 128 — Constructive

(37, 48, 1775641)-net in base 128, using

base change [i] based on digital (31, 42, 1775641)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 11, 97921)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256¹¹, 97921, F₂₅₆, 5, 5) (dual of [(97921, 5), 489594, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(256¹¹, 195843, F₂₅₆, 5) (dual of [195843, 195832, 6]-code), using
        (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
        linear OA(256¹, 65281, F₂₅₆, 1) (dual of [65281, 65280, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OA(256³, 65281, F₂₅₆, 2) (dual of [65281, 65278, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(256³, 65793, F₂₅₆, 2) (dual of [65793, 65790, 3]-code), using
        Hamming code H(3,256) [i]
        linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (20, 31, 1677720)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256³¹, 1677720, F₂₅₆, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(256³¹, 8388601, F₂₅₆, 11) (dual of [8388601, 8388570, 12]-code), using
        discarding factors / shortening the dual code 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]

(37, 48, large)-Net over F₁₂₈ — Digital

Digital (37, 48, large)-net over F₁₂₈, using

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

(37, 48, large)-Net in Base 128 — Upper bound on s

There is no (37, 48, large)-net in base 128, because

9 times m-reduction [i] would yield (37, 39, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (37, 48, s)-Nets in Base 128

(37, 48, 1685848)-Net over F128 — Constructive and digital

(37, 48, 1775641)-Net in Base 128 — Constructive

(37, 48, large)-Net over F128 — Digital

(37, 48, large)-Net in Base 128 — Upper bound on s

(37, 48, 1685848)-Net over F₁₂₈ — Constructive and digital

(37, 48, large)-Net over F₁₂₈ — Digital