MinT - Best Known (42−17, 42, s)-Nets in Base 256

Best Known (42−17, 42, s)-Nets in Base 256

(42−17, 42, 8450)-Net over F₂₅₆ — Constructive and digital

Digital (25, 42, 8450)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (16, 33, 8192)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256³³, 8192, F₂₅₆, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(256³³, 65537, F₂₅₆, 17) (dual of [65537, 65504, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(42−17, 42, 65827)-Net over F₂₅₆ — Digital

Digital (25, 42, 65827)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁴², 65827, F₂₅₆, 17) (dual of [65827, 65785, 18]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(256⁹, 289, F₂₅₆, 8) (dual of [289, 280, 9]-code), using
    - extended algebraic-geometric code AG_e(F,280P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]
  2. linear OA(256³³, 65538, F₂₅₆, 17) (dual of [65538, 65505, 18]-code), using
    - constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
      1. 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]
      2. linear OA(256³¹, 65536, F₂₅₆, 16) (dual of [65536, 65505, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. 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]

(42−17, 42, large)-Net in Base 256 — Upper bound on s

There is no (25, 42, large)-net in base 256, because

15 times m-reduction [i] would yield (25, 27, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]

Best Known (42−17, 42, s)-Nets in Base 256

(42−17, 42, 8450)-Net over F256 — Constructive and digital

(42−17, 42, 65827)-Net over F256 — Digital

(42−17, 42, large)-Net in Base 256 — Upper bound on s

(42−17, 42, 8450)-Net over F₂₅₆ — Constructive and digital

(42−17, 42, 65827)-Net over F₂₅₆ — Digital