Best Known (178−19, 178, s)-Nets in Base 2
(178−19, 178, 58257)-Net over F2 — Constructive and digital
Digital (159, 178, 58257)-net over F2, using
- net defined by OOA [i] based on linear OOA(2178, 58257, F2, 19, 19) (dual of [(58257, 19), 1106705, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2178, 524314, F2, 19) (dual of [524314, 524136, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 524320, F2, 19) (dual of [524320, 524142, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2178, 524320, F2, 19) (dual of [524320, 524142, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2178, 524314, F2, 19) (dual of [524314, 524136, 20]-code), using
(178−19, 178, 87386)-Net over F2 — Digital
Digital (159, 178, 87386)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2178, 87386, F2, 6, 19) (dual of [(87386, 6), 524138, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2178, 524316, F2, 19) (dual of [524316, 524138, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 524320, F2, 19) (dual of [524320, 524142, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2178, 524320, F2, 19) (dual of [524320, 524142, 20]-code), using
- OOA 6-folding [i] based on linear OA(2178, 524316, F2, 19) (dual of [524316, 524138, 20]-code), using
(178−19, 178, 3451488)-Net in Base 2 — Upper bound on s
There is no (159, 178, 3451489)-net in base 2, because
- 1 times m-reduction [i] would yield (159, 177, 3451489)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 191562 339142 758771 466129 597304 224959 140055 220383 131848 > 2177 [i]