MinT - Best Known (215, 215+39, s)-Nets in Base 2

Best Known (215, 215+39, s)-Nets in Base 2

(215, 215+39, 624)-Net over F₂ — Constructive and digital

Digital (215, 254, 624)-net over F₂, using

2² times duplication [i] based on digital (213, 252, 624)-net over F₂, using
- trace code for nets [i] based on digital (3, 42, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(215, 215+39, 2056)-Net over F₂ — Digital

Digital (215, 254, 2056)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁵⁴, 2056, F₂, 4, 39) (dual of [(2056, 4), 7970, 40]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2²⁵⁴, 8224, F₂, 39) (dual of [8224, 7970, 40]-code), using
  - constructionÂ X applied to C_e(38) âŠ‚ C_e(34) [i] based on
    1. linear OA(2²⁴⁸, 8192, F₂, 39) (dual of [8192, 7944, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    2. linear OA(2²²², 8192, F₂, 35) (dual of [8192, 7970, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(2⁶, 32, F₂, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
      - Reedâ€“Muller code RM(1,5) [i]
      - caps in base bÂ =Â 2 [i]

(215, 215+39, 80819)-Net in Base 2 — Upper bound on s

There is no (215, 254, 80820)-net in base 2, because

1 times m-reduction [i] would yield (215, 253, 80820)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14475 089750 658262 952231 079447 231041 010631 012158 431050 125484 568540 690517 721273 > 2²⁵³ [i]