Best Known (45, 54, s)-Nets in Base 2
(45, 54, 2051)-Net over F2 — Constructive and digital
Digital (45, 54, 2051)-net over F2, using
- net defined by OOA [i] based on linear OOA(254, 2051, F2, 9, 9) (dual of [(2051, 9), 18405, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(254, 2051, F2, 8, 9) (dual of [(2051, 8), 16354, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(254, 8205, F2, 9) (dual of [8205, 8151, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(254, 8206, F2, 9) (dual of [8206, 8152, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(253, 8192, F2, 9) (dual of [8192, 8139, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(240, 8192, F2, 7) (dual of [8192, 8152, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(254, 8206, F2, 9) (dual of [8206, 8152, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(254, 8205, F2, 9) (dual of [8205, 8151, 10]-code), using
- appending kth column [i] based on linear OOA(254, 2051, F2, 8, 9) (dual of [(2051, 8), 16354, 10]-NRT-code), using
(45, 54, 2735)-Net over F2 — Digital
Digital (45, 54, 2735)-net over F2, using
- net defined by OOA [i] based on linear OOA(254, 2735, F2, 9, 9) (dual of [(2735, 9), 24561, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(254, 2735, F2, 8, 9) (dual of [(2735, 8), 21826, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(254, 2735, F2, 3, 9) (dual of [(2735, 3), 8151, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(254, 8205, F2, 9) (dual of [8205, 8151, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(254, 8206, F2, 9) (dual of [8206, 8152, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(253, 8192, F2, 9) (dual of [8192, 8139, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(240, 8192, F2, 7) (dual of [8192, 8152, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(254, 8206, F2, 9) (dual of [8206, 8152, 10]-code), using
- OOA 3-folding [i] based on linear OA(254, 8205, F2, 9) (dual of [8205, 8151, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(254, 2735, F2, 3, 9) (dual of [(2735, 3), 8151, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(254, 2735, F2, 8, 9) (dual of [(2735, 8), 21826, 10]-NRT-code), using
(45, 54, 21557)-Net in Base 2 — Upper bound on s
There is no (45, 54, 21558)-net in base 2, because
- 1 times m-reduction [i] would yield (45, 53, 21558)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 9008 774656 337609 > 253 [i]