MinT - Best Known (127−13, 127, s)-Nets in Base 2

Best Known (127−13, 127, s)-Nets in Base 2

Digital (114, 127, 349525)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹²⁷, 349525, F₂, 13, 13) (dual of [(349525, 13), 4543698, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2¹²⁷, 2097151, F₂, 13) (dual of [2097151, 2097024, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹²⁷, 2097152, F₂, 13) (dual of [2097152, 2097025, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 2²¹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

Digital (114, 127, 419430)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹²⁷, 419430, F₂, 5, 13) (dual of [(419430, 5), 2097023, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2¹²⁷, 2097150, F₂, 13) (dual of [2097150, 2097023, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹²⁷, 2097152, F₂, 13) (dual of [2097152, 2097025, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 2²¹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

There is no (114, 127, 6278436)-net in base 2, because

1 times m-reduction [i] would yield (114, 126, 6278436)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 85 070671 485522 119560 251172 936060 117288 > 2¹²⁶ [i]