MinT - Best Known (50−18, 50, s)-Nets in Base 256

Best Known (50−18, 50, s)-Nets in Base 256

(50−18, 50, 7798)-Net over F₂₅₆ — Constructive and digital

Digital (32, 50, 7798)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 15, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (1, 10, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)
2. digital (17, 35, 7282)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256³⁵, 7282, F₂₅₆, 18, 18) (dual of [(7282, 18), 131041, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(256³⁵, 65538, F₂₅₆, 18) (dual of [65538, 65503, 19]-code), using
      - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
        linear OA(256³⁵, 65536, F₂₅₆, 18) (dual of [65536, 65501, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(256³³, 65536, F₂₅₆, 17) (dual of [65536, 65503, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(50−18, 50, 340780)-Net over F₂₅₆ — Digital

Digital (32, 50, 340780)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−18, 50, large)-Net in Base 256 — Upper bound on s

There is no (32, 50, large)-net in base 256, because

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

Best Known (50−18, 50, s)-Nets in Base 256

(50−18, 50, 7798)-Net over F256 — Constructive and digital

(50−18, 50, 340780)-Net over F256 — Digital

(50−18, 50, large)-Net in Base 256 — Upper bound on s

(50−18, 50, 7798)-Net over F₂₅₆ — Constructive and digital

(50−18, 50, 340780)-Net over F₂₅₆ — Digital