Best Known (260−32, 260, s)-Nets in Base 2
(260−32, 260, 4097)-Net over F2 — Constructive and digital
Digital (228, 260, 4097)-net over F2, using
- 22 times duplication [i] based on digital (226, 258, 4097)-net over F2, using
- t-expansion [i] based on digital (225, 258, 4097)-net over F2, using
- net defined by OOA [i] based on linear OOA(2258, 4097, F2, 33, 33) (dual of [(4097, 33), 134943, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2257, 65536, F2, 33) (dual of [65536, 65279, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(32) ⊂ Ce(30) [i] based on
- OOA 16-folding and stacking with additional row [i] based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- net defined by OOA [i] based on linear OOA(2258, 4097, F2, 33, 33) (dual of [(4097, 33), 134943, 34]-NRT-code), using
- t-expansion [i] based on digital (225, 258, 4097)-net over F2, using
(260−32, 260, 10925)-Net over F2 — Digital
Digital (228, 260, 10925)-net over F2, using
- 22 times duplication [i] based on digital (226, 258, 10925)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 10925, F2, 6, 32) (dual of [(10925, 6), 65292, 33]-NRT-code), using
- strength reduction [i] based on linear OOA(2258, 10925, F2, 6, 33) (dual of [(10925, 6), 65292, 34]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2258, 65550, F2, 33) (dual of [65550, 65292, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2257, 65536, F2, 33) (dual of [65536, 65279, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(32) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- OOA 6-folding [i] based on linear OA(2258, 65550, F2, 33) (dual of [65550, 65292, 34]-code), using
- strength reduction [i] based on linear OOA(2258, 10925, F2, 6, 33) (dual of [(10925, 6), 65292, 34]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 10925, F2, 6, 32) (dual of [(10925, 6), 65292, 33]-NRT-code), using
(260−32, 260, 529976)-Net in Base 2 — Upper bound on s
There is no (228, 260, 529977)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 852681 944266 933614 106062 774439 333235 494602 149977 838930 988661 223899 209726 607209 > 2260 [i]