Best Known (7, 7+4, s)-Nets in Base 2
(7, 7+4, 47)-Net over F2 — Constructive and digital
Digital (7, 11, 47)-net over F2, using
(7, 7+4, 57)-Net over F2 — Upper bound on s (digital)
There is no digital (7, 11, 58)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(211, 58, F2, 4) (dual of [58, 47, 5]-code), but
- construction Y1 [i] would yield
- linear OA(210, 34, F2, 4) (dual of [34, 24, 5]-code), but
- “BoV†bound on codes from Brouwer’s database [i]
- linear OA(247, 58, F2, 24) (dual of [58, 11, 25]-code), but
- discarding factors / shortening the dual code would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- “vT4†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- discarding factors / shortening the dual code would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- linear OA(210, 34, F2, 4) (dual of [34, 24, 5]-code), but
- construction Y1 [i] would yield
(7, 7+4, 61)-Net in Base 2 — Upper bound on s
There is no (7, 11, 62)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2078 > 211 [i]