MinT - Best Known (260−41, 260, s)-Nets in Base 2

Best Known (260−41, 260, s)-Nets in Base 2

(260−41, 260, 600)-Net over F₂ — Constructive and digital

Digital (219, 260, 600)-net over F₂, using

trace code for nets [i] based on digital (11, 52, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(260−41, 260, 1607)-Net over F₂ — Digital

Digital (219, 260, 1607)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁶⁰, 1607, F₂, 2, 41) (dual of [(1607, 2), 2954, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁶⁰, 2081, F₂, 2, 41) (dual of [(2081, 2), 3902, 42]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²⁶⁰, 4162, F₂, 41) (dual of [4162, 3902, 42]-code), using
    - constructionÂ X applied to C([0,20]) âŠ‚ C([0,16]) [i] based on
      1. linear OA(2²⁴¹, 4097, F₂, 41) (dual of [4097, 3856, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 2²⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
      2. linear OA(2¹⁹³, 4097, F₂, 33) (dual of [4097, 3904, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 2²⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      3. linear OA(2¹⁹, 65, F₂, 7) (dual of [65, 46, 8]-code), using
        adding a parity check bit [i] based on linear OA(2¹⁸, 64, F₂, 6) (dual of [64, 46, 7]-code), using
        extracting embedded orthogonal array [i] based on digital (12, 18, 64)-net over F₂, using
        linear code embedding found in a computer search [i]

(260−41, 260, 65682)-Net in Base 2 — Upper bound on s

There is no (219, 260, 65683)-net in base 2, because

1 times m-reduction [i] would yield (219, 259, 65683)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 926476 453285 481582 919813 940921 392374 858169 412946 809051 983509 001728 859514 394728 > 2²⁵⁹ [i]