MinT - Best Known (137−11, 137, s)-Nets in Base 2

Best Known (137−11, 137, s)-Nets in Base 2

(137−11, 137, 2097151)-Net over F₂ — Constructive and digital

Digital (126, 137, 2097151)-net over F₂, using

2³ times duplication [i] based on digital (123, 134, 2097151)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹³⁴, 2097151, F₂, 15, 11) (dual of [(2097151, 15), 31457131, 12]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹³⁴, 4194303, F₂, 3, 11) (dual of [(4194303, 3), 12582775, 12]-NRT-code), using
    - OOAs constructed from BCH codes [i]

(137−11, 137, 2097692)-Net over F₂ — Digital

Digital (126, 137, 2097692)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹³⁷, 2097692, F₂, 4, 11) (dual of [(2097692, 4), 8390631, 12]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²¹, 542, F₂, 4, 5) (dual of [(542, 4), 2147, 6]-NRT-code), using
    - extracting embedded OOA [i] based on digital (16, 21, 542)-net over F₂, using
      - linear code embedding found in a computer search [i]
  2. linear OOA(2¹¹⁶, 2097150, F₂, 4, 11) (dual of [(2097150, 4), 8388484, 12]-NRT-code), using
    - OOA 4-folding [i] based on linear OA(2¹¹⁶, 8388600, F₂, 11) (dual of [8388600, 8388484, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹¹⁶, large, F₂, 11) (dual of [large, large−116, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(137−11, 137, large)-Net in Base 2 — Upper bound on s

There is no (126, 137, large)-net in base 2, because

9 times m-reduction [i] would yield (126, 128, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]