MinT - Best Known (72−17, 72, s)-Nets in Base 128

Best Known (72−17, 72, s)-Nets in Base 128

(72−17, 72, 1048575)-Net over F₁₂₈ — Constructive and digital

Digital (55, 72, 1048575)-net over F₁₂₈, using

128⁷ times duplication [i] based on digital (48, 65, 1048575)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128⁶⁵, 1048575, F₁₂₈, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
  - OOA 8-folding and stacking with additional row [i] based on linear OA(128⁶⁵, 8388601, F₁₂₈, 17) (dual of [8388601, 8388536, 18]-code), using
    - discarding factors / shortening the dual code based on linear OA(128⁶⁵, large, F₁₂₈, 17) (dual of [large, large−65, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 15790321Â | 128⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(72−17, 72, 1052672)-Net in Base 128 — Constructive

(55, 72, 1052672)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (8, 16, 4097)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128¹⁶, 4097, F₁₂₈, 8, 8) (dual of [(4097, 8), 32760, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(128¹⁶, 16388, F₁₂₈, 8) (dual of [16388, 16372, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(128¹⁶, 16389, F₁₂₈, 8) (dual of [16389, 16373, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(5) [i] based on
        linear OA(128¹⁵, 16384, F₁₂₈, 8) (dual of [16384, 16369, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(128¹¹, 16384, F₁₂₈, 6) (dual of [16384, 16373, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(128¹, 5, F₁₂₈, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. (39, 56, 1048575)-net in base 128, using
  - base change [i] based on digital (32, 49, 1048575)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁹, 1048575, F₂₅₆, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
      - OOA 8-folding and stacking with additional row [i] based on linear OA(256⁴⁹, 8388601, F₂₅₆, 17) (dual of [8388601, 8388552, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(72−17, 72, large)-Net over F₁₂₈ — Digital

Digital (55, 72, large)-net over F₁₂₈, using

t-expansion [i] based on digital (54, 72, large)-net over F₁₂₈, using
- 2 times m-reduction [i] based on digital (54, 74, large)-net over F₁₂₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(72−17, 72, large)-Net in Base 128 — Upper bound on s

There is no (55, 72, large)-net in base 128, because

15 times m-reduction [i] would yield (55, 57, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (72−17, 72, s)-Nets in Base 128

(72−17, 72, 1048575)-Net over F128 — Constructive and digital

(72−17, 72, 1052672)-Net in Base 128 — Constructive

(72−17, 72, large)-Net over F128 — Digital

(72−17, 72, large)-Net in Base 128 — Upper bound on s

(72−17, 72, 1048575)-Net over F₁₂₈ — Constructive and digital

(72−17, 72, large)-Net over F₁₂₈ — Digital