MinT - Best Known (13, 25, s)-Nets in Base 2

Best Known (13, 25, s)-Nets in Base 2

(13, 25, 25)-Net over F₂ — Constructive and digital

Digital (13, 25, 25)-net over F₂, using

a shift-net [i]

(13, 25, 37)-Net over F₂ — Upper bound on s (digital)

There is no digital (13, 25, 38)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁵, 38, F₂, 12) (dual of [38, 13, 13]-code), but
- adding a parity check bit [i] would yield linear OA(2²⁶, 39, F₂, 13) (dual of [39, 13, 14]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(13, 25, 45)-Net in Base 2 — Upper bound on s

There is no (13, 25, 46)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 34 262572 > 2²⁵ [i]