Best Known (101−17, 101, s)-Nets in Base 2
(101−17, 101, 513)-Net over F2 — Constructive and digital
Digital (84, 101, 513)-net over F2, using
- 23 times duplication [i] based on digital (81, 98, 513)-net over F2, using
- net defined by OOA [i] based on linear OOA(298, 513, F2, 17, 17) (dual of [(513, 17), 8623, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(298, 4105, F2, 17) (dual of [4105, 4007, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(298, 4109, F2, 17) (dual of [4109, 4011, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(297, 4096, F2, 17) (dual of [4096, 3999, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(285, 4096, F2, 15) (dual of [4096, 4011, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(298, 4109, F2, 17) (dual of [4109, 4011, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(298, 4105, F2, 17) (dual of [4105, 4007, 18]-code), using
- net defined by OOA [i] based on linear OOA(298, 513, F2, 17, 17) (dual of [(513, 17), 8623, 18]-NRT-code), using
(101−17, 101, 1035)-Net over F2 — Digital
Digital (84, 101, 1035)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2101, 1035, F2, 3, 17) (dual of [(1035, 3), 3004, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2101, 1370, F2, 3, 17) (dual of [(1370, 3), 4009, 18]-NRT-code), using
- 22 times duplication [i] based on linear OOA(299, 1370, F2, 3, 17) (dual of [(1370, 3), 4011, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(299, 4110, F2, 17) (dual of [4110, 4011, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(298, 4109, F2, 17) (dual of [4109, 4011, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(297, 4096, F2, 17) (dual of [4096, 3999, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(285, 4096, F2, 15) (dual of [4096, 4011, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(16) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(298, 4109, F2, 17) (dual of [4109, 4011, 18]-code), using
- OOA 3-folding [i] based on linear OA(299, 4110, F2, 17) (dual of [4110, 4011, 18]-code), using
- 22 times duplication [i] based on linear OOA(299, 1370, F2, 3, 17) (dual of [(1370, 3), 4011, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2101, 1370, F2, 3, 17) (dual of [(1370, 3), 4009, 18]-NRT-code), using
(101−17, 101, 21793)-Net in Base 2 — Upper bound on s
There is no (84, 101, 21794)-net in base 2, because
- 1 times m-reduction [i] would yield (84, 100, 21794)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 267673 594887 340946 638444 849500 > 2100 [i]