MinT - Best Known (43−11, 43, s)-Nets in Base 256

Best Known (43−11, 43, s)-Nets in Base 256

(43−11, 43, 1808281)-Net over F₂₅₆ — Constructive and digital

Digital (32, 43, 1808281)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (7, 12, 130561)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256¹², 130561, F₂₅₆, 5, 5) (dual of [(130561, 5), 652793, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(256¹², 261123, F₂₅₆, 5) (dual of [261123, 261111, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(256¹², 261124, F₂₅₆, 5) (dual of [261124, 261112, 6]-code), using
        generalized (u,Â u+v)-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(256¹, 65281, F₂₅₆, 1) (dual of [65281, 65280, 2]-code) (see above)
        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]

(43−11, 43, large)-Net over F₂₅₆ — Digital

Digital (32, 43, large)-net over F₂₅₆, using

t-expansion [i] based on digital (30, 43, large)-net over F₂₅₆, using
- 1 times m-reduction [i] based on digital (30, 44, large)-net over F₂₅₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁴⁴, large, F₂₅₆, 14) (dual of [large, large−44, 15]-code), using
    - 4 times code embedding in larger space [i] based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(43−11, 43, large)-Net in Base 256 — Upper bound on s

There is no (32, 43, large)-net in base 256, because

9 times m-reduction [i] would yield (32, 34, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]

Best Known (43−11, 43, s)-Nets in Base 256

(43−11, 43, 1808281)-Net over F256 — Constructive and digital

(43−11, 43, large)-Net over F256 — Digital

(43−11, 43, large)-Net in Base 256 — Upper bound on s

(43−11, 43, 1808281)-Net over F₂₅₆ — Constructive and digital

(43−11, 43, large)-Net over F₂₅₆ — Digital