Best Known (172, 172+21, s)-Nets in Base 2
(172, 172+21, 52430)-Net over F2 — Constructive and digital
Digital (172, 193, 52430)-net over F2, using
- 21 times duplication [i] based on digital (171, 192, 52430)-net over F2, using
- net defined by OOA [i] based on linear OOA(2192, 52430, F2, 21, 21) (dual of [(52430, 21), 1100838, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2192, 524301, F2, 21) (dual of [524301, 524109, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2192, 524308, F2, 21) (dual of [524308, 524116, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2191, 524288, F2, 21) (dual of [524288, 524097, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(21, 20, F2, 1) (dual of [20, 19, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2192, 524308, F2, 21) (dual of [524308, 524116, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2192, 524301, F2, 21) (dual of [524301, 524109, 22]-code), using
- net defined by OOA [i] based on linear OOA(2192, 52430, F2, 21, 21) (dual of [(52430, 21), 1100838, 22]-NRT-code), using
(172, 172+21, 74901)-Net over F2 — Digital
Digital (172, 193, 74901)-net over F2, using
- 21 times duplication [i] based on digital (171, 192, 74901)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2192, 74901, F2, 7, 21) (dual of [(74901, 7), 524115, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2192, 524307, F2, 21) (dual of [524307, 524115, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2192, 524308, F2, 21) (dual of [524308, 524116, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2191, 524288, F2, 21) (dual of [524288, 524097, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(21, 20, F2, 1) (dual of [20, 19, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2192, 524308, F2, 21) (dual of [524308, 524116, 22]-code), using
- OOA 7-folding [i] based on linear OA(2192, 524307, F2, 21) (dual of [524307, 524115, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2192, 74901, F2, 7, 21) (dual of [(74901, 7), 524115, 22]-NRT-code), using
(172, 172+21, 2727406)-Net in Base 2 — Upper bound on s
There is no (172, 193, 2727407)-net in base 2, because
- 1 times m-reduction [i] would yield (172, 192, 2727407)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6277 107464 444004 213678 322922 311278 678879 165509 610685 070673 > 2192 [i]