Best Known (54, 54+11, s)-Nets in Base 2
(54, 54+11, 822)-Net over F2 — Constructive and digital
Digital (54, 65, 822)-net over F2, using
- 22 times duplication [i] based on digital (52, 63, 822)-net over F2, using
- net defined by OOA [i] based on linear OOA(263, 822, F2, 11, 11) (dual of [(822, 11), 8979, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(263, 4111, F2, 11) (dual of [4111, 4048, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(262, 4110, F2, 11) (dual of [4110, 4048, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(261, 4096, F2, 11) (dual of [4096, 4035, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [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 X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(262, 4110, F2, 11) (dual of [4110, 4048, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(263, 4111, F2, 11) (dual of [4111, 4048, 12]-code), using
- net defined by OOA [i] based on linear OOA(263, 822, F2, 11, 11) (dual of [(822, 11), 8979, 12]-NRT-code), using
(54, 54+11, 1371)-Net over F2 — Digital
Digital (54, 65, 1371)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(265, 1371, F2, 3, 11) (dual of [(1371, 3), 4048, 12]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(262, 1370, F2, 3, 11) (dual of [(1370, 3), 4048, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(262, 4110, F2, 11) (dual of [4110, 4048, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(261, 4096, F2, 11) (dual of [4096, 4035, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [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 X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- OOA 3-folding [i] based on linear OA(262, 4110, F2, 11) (dual of [4110, 4048, 12]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(262, 1370, F2, 3, 11) (dual of [(1370, 3), 4048, 12]-NRT-code), using
(54, 54+11, 18571)-Net in Base 2 — Upper bound on s
There is no (54, 65, 18572)-net in base 2, because
- 1 times m-reduction [i] would yield (54, 64, 18572)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 18 447193 170127 683546 > 264 [i]