Best Known (178−23, 178, s)-Nets in Base 2
(178−23, 178, 5959)-Net over F2 — Constructive and digital
Digital (155, 178, 5959)-net over F2, using
- net defined by OOA [i] based on linear OOA(2178, 5959, F2, 23, 23) (dual of [(5959, 23), 136879, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2178, 65550, F2, 23) (dual of [65550, 65372, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2161, 65536, F2, 21) (dual of [65536, 65375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2178, 65550, F2, 23) (dual of [65550, 65372, 24]-code), using
(178−23, 178, 10925)-Net over F2 — Digital
Digital (155, 178, 10925)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2178, 10925, F2, 6, 23) (dual of [(10925, 6), 65372, 24]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2178, 65550, F2, 23) (dual of [65550, 65372, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2161, 65536, F2, 21) (dual of [65536, 65375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- OOA 6-folding [i] based on linear OA(2178, 65550, F2, 23) (dual of [65550, 65372, 24]-code), using
(178−23, 178, 342641)-Net in Base 2 — Upper bound on s
There is no (155, 178, 342642)-net in base 2, because
- 1 times m-reduction [i] would yield (155, 177, 342642)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 191564 005785 100826 590699 199788 693972 228154 671604 183712 > 2177 [i]