MinT - Best Known (105, 113, s)-Nets in Base 2

Best Known (105, 113, s)-Nets in Base 2

(105, 113, 2098175)-Net over F₂ — Constructive and digital

Digital (105, 113, 2098175)-net over F₂, using

t-expansion [i] based on digital (104, 113, 2098175)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹¹³, 2098175, F₂, 9, 9) (dual of [(2098175, 9), 18883462, 10]-NRT-code), using
  - appending kth column [i] based on linear OOA(2¹¹³, 2098175, F₂, 8, 9) (dual of [(2098175, 8), 16785287, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²⁰, 1025, F₂, 8, 4) (dual of [(1025, 8), 8180, 5]-NRT-code), using
        appending 4 arbitrary columns [i] based on linear OOA(2²⁰, 1025, F₂, 4, 4) (dual of [(1025, 4), 4080, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2²⁰, 1025, F₂, 3, 4) (dual of [(1025, 3), 3055, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (16, 20, 1025)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
      2. linear OOA(2⁹³, 2097150, F₂, 8, 9) (dual of [(2097150, 8), 16777107, 10]-NRT-code), using
        OOA 4-folding and stacking with additional row [i] based on linear OA(2⁹³, 8388601, F₂, 9) (dual of [8388601, 8388508, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(2⁹³, large, F₂, 9) (dual of [large, large−93, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(105, 113, 4195336)-Net over F₂ — Digital

Digital (105, 113, 4195336)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹¹³, 4195336, F₂, 8, 8) (dual of [(4195336, 8), 33562575, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹¹³, 4195336, F₂, 7, 8) (dual of [(4195336, 7), 29367239, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹³, 4195336, F₂, 2, 8) (dual of [(4195336, 2), 8390559, 9]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²¹, 1035, F₂, 2, 4) (dual of [(1035, 2), 2049, 5]-NRT-code), using
        embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(2²¹, 1035, F₂, 4) (dual of [1035, 1014, 5]-code), using
        1 times truncation [i] based on linear OA(2²², 1036, F₂, 5) (dual of [1036, 1014, 6]-code), using
        constructionÂ X4 applied to C_e(4) âŠ‚ C_e(2) [i] based on
        linear OA(2²¹, 1024, F₂, 5) (dual of [1024, 1003, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(2¹¹, 1024, F₂, 3) (dual of [1024, 1013, 4]-code or 1024-cap in PG(10,2)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(2¹¹, 12, F₂, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,2)), using
        dual of repetition code with length 12 [i]
        linear OA(2¹, 12, F₂, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
      2. linear OOA(2⁹², 4194301, F₂, 2, 8) (dual of [(4194301, 2), 8388510, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2⁹², 8388602, F₂, 8) (dual of [8388602, 8388510, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(2⁹², large, F₂, 8) (dual of [large, large−92, 9]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(105, 113, large)-Net in Base 2 — Upper bound on s

There is no (105, 113, large)-net in base 2, because

6 times m-reduction [i] would yield (105, 107, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (105, 113, s)-Nets in Base 2

(105, 113, 2098175)-Net over F2 — Constructive and digital

(105, 113, 4195336)-Net over F2 — Digital

(105, 113, large)-Net in Base 2 — Upper bound on s

(105, 113, 2098175)-Net over F₂ — Constructive and digital

(105, 113, 4195336)-Net over F₂ — Digital