Best Known (174−22, 174, s)-Nets in Base 2
(174−22, 174, 2982)-Net over F2 — Constructive and digital
Digital (152, 174, 2982)-net over F2, using
- 21 times duplication [i] based on digital (151, 173, 2982)-net over F2, using
- t-expansion [i] based on digital (150, 173, 2982)-net over F2, using
- net defined by OOA [i] based on linear OOA(2173, 2982, F2, 23, 23) (dual of [(2982, 23), 68413, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2173, 32803, F2, 23) (dual of [32803, 32630, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2173, 32805, F2, 23) (dual of [32805, 32632, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2136, 32768, F2, 19) (dual of [32768, 32632, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 37, F2, 3) (dual of [37, 30, 4]-code or 37-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2173, 32805, F2, 23) (dual of [32805, 32632, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2173, 32803, F2, 23) (dual of [32803, 32630, 24]-code), using
- net defined by OOA [i] based on linear OOA(2173, 2982, F2, 23, 23) (dual of [(2982, 23), 68413, 24]-NRT-code), using
- t-expansion [i] based on digital (150, 173, 2982)-net over F2, using
(174−22, 174, 7325)-Net over F2 — Digital
Digital (152, 174, 7325)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2174, 7325, F2, 4, 22) (dual of [(7325, 4), 29126, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2174, 8201, F2, 4, 22) (dual of [(8201, 4), 32630, 23]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2172, 8201, F2, 4, 22) (dual of [(8201, 4), 32632, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2172, 32804, F2, 22) (dual of [32804, 32632, 23]-code), using
- 1 times truncation [i] based on linear OA(2173, 32805, F2, 23) (dual of [32805, 32632, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2136, 32768, F2, 19) (dual of [32768, 32632, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 37, F2, 3) (dual of [37, 30, 4]-code or 37-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2173, 32805, F2, 23) (dual of [32805, 32632, 24]-code), using
- OOA 4-folding [i] based on linear OA(2172, 32804, F2, 22) (dual of [32804, 32632, 23]-code), using
- 22 times duplication [i] based on linear OOA(2172, 8201, F2, 4, 22) (dual of [(8201, 4), 32632, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2174, 8201, F2, 4, 22) (dual of [(8201, 4), 32630, 23]-NRT-code), using
(174−22, 174, 283620)-Net in Base 2 — Upper bound on s
There is no (152, 174, 283621)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 23946 166107 543386 368913 151382 878393 327263 267198 552832 > 2174 [i]