MinT - Best Known (121−10, 121, s)-Nets in Base 2

Best Known (121−10, 121, s)-Nets in Base 2

(121−10, 121, 1677725)-Net over F₂ — Constructive and digital

Digital (111, 121, 1677725)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - nets constructed from net-embeddable BCH codes [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (105, 115, 1677720)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹¹⁵, 1677720, F₂, 10, 10) (dual of [(1677720, 10), 16777085, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(2¹¹⁵, 8388600, F₂, 10) (dual of [8388600, 8388485, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹¹⁵, large, F₂, 10) (dual of [large, large−115, 11]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(121−10, 121, 2491594)-Net over F₂ — Digital

Digital (111, 121, 2491594)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹²¹, 2491594, F₂, 3, 10) (dual of [(2491594, 3), 7474661, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹²¹, 2796206, F₂, 3, 10) (dual of [(2796206, 3), 8388497, 11]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁶, 5, F₂, 3, 5) (dual of [(5, 3), 9, 6]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,9P) [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
        an explicitly constructive algebraic function field [i]
    2. linear OOA(2¹¹⁵, 2796201, F₂, 3, 10) (dual of [(2796201, 3), 8388488, 11]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(2¹¹⁵, large, F₂, 10) (dual of [large, large−115, 11]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(121−10, 121, large)-Net in Base 2 — Upper bound on s

There is no (111, 121, large)-net in base 2, because

8 times m-reduction [i] would yield (111, 113, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]