Best Known (53−8, 53, s)-Nets in Base 2
(53−8, 53, 2051)-Net over F2 — Constructive and digital
Digital (45, 53, 2051)-net over F2, using
- net defined by OOA [i] based on linear OOA(253, 2051, F2, 8, 8) (dual of [(2051, 8), 16355, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(253, 8204, F2, 8) (dual of [8204, 8151, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(253, 8206, F2, 8) (dual of [8206, 8153, 9]-code), using
- 1 times truncation [i] based on linear OA(254, 8207, F2, 9) (dual of [8207, 8153, 10]-code), using
- construction X4 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(214, 15, F2, 13) (dual of [15, 1, 14]-code), using
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- dual of repetition code with length 15 [i]
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(254, 8207, F2, 9) (dual of [8207, 8153, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(253, 8206, F2, 8) (dual of [8206, 8153, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(253, 8204, F2, 8) (dual of [8204, 8151, 9]-code), using
(53−8, 53, 3058)-Net over F2 — Digital
Digital (45, 53, 3058)-net over F2, using
- net defined by OOA [i] based on linear OOA(253, 3058, F2, 8, 8) (dual of [(3058, 8), 24411, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(253, 3058, F2, 7, 8) (dual of [(3058, 7), 21353, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(253, 3058, F2, 2, 8) (dual of [(3058, 2), 6063, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(253, 4103, F2, 2, 8) (dual of [(4103, 2), 8153, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(253, 8206, F2, 8) (dual of [8206, 8153, 9]-code), using
- 1 times truncation [i] based on linear OA(254, 8207, F2, 9) (dual of [8207, 8153, 10]-code), using
- construction X4 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(214, 15, F2, 13) (dual of [15, 1, 14]-code), using
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- dual of repetition code with length 15 [i]
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(254, 8207, F2, 9) (dual of [8207, 8153, 10]-code), using
- OOA 2-folding [i] based on linear OA(253, 8206, F2, 8) (dual of [8206, 8153, 9]-code), using
- discarding factors / shortening the dual code based on linear OOA(253, 4103, F2, 2, 8) (dual of [(4103, 2), 8153, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(253, 3058, F2, 2, 8) (dual of [(3058, 2), 6063, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(253, 3058, F2, 7, 8) (dual of [(3058, 7), 21353, 9]-NRT-code), using
(53−8, 53, 21557)-Net in Base 2 — Upper bound on s
There is no (45, 53, 21558)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 9008 774656 337609 > 253 [i]