Best Known (81, 81+17, s)-Nets in Base 2
(81, 81+17, 513)-Net over F2 — Constructive and digital
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
(81, 81+17, 1027)-Net over F2 — Digital
Digital (81, 98, 1027)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(298, 1027, F2, 4, 17) (dual of [(1027, 4), 4010, 18]-NRT-code), using
- OOA 4-folding [i] based on linear OA(298, 4108, F2, 17) (dual of [4108, 4010, 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 4-folding [i] based on linear OA(298, 4108, F2, 17) (dual of [4108, 4010, 18]-code), using
(81, 81+17, 16802)-Net in Base 2 — Upper bound on s
There is no (81, 98, 16803)-net in base 2, because
- 1 times m-reduction [i] would yield (81, 97, 16803)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 158471 225643 286069 172112 178876 > 297 [i]