MinT - Best Known (15−6, 15, s)-Nets in Base 2

Best Known (15−6, 15, s)-Nets in Base 2

(15−6, 15, 36)-Net over F₂ — Constructive and digital

Digital (9, 15, 36)-net over F₂, using

linear code embedding found in a computer search [i]

(15−6, 15, 53)-Net over F₂ — Upper bound on s (digital)

There is no digital (9, 15, 54)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁵, 54, F₂, 6) (dual of [54, 39, 7]-code), but
- adding a parity check bit [i] would yield linear OA(2¹⁶, 55, F₂, 7) (dual of [55, 39, 8]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(15−6, 15, 54)-Net in Base 2 — Upper bound on s

There is no (9, 15, 55)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 34046 > 2¹⁵ [i]