Best Known (167, 167+17, s)-Nets in Base 2
(167, 167+17, 524294)-Net over F2 — Constructive and digital
Digital (167, 184, 524294)-net over F2, using
- net defined by OOA [i] based on linear OOA(2184, 524294, F2, 17, 17) (dual of [(524294, 17), 8912814, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2184, 4194353, F2, 17) (dual of [4194353, 4194169, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2184, 4194356, F2, 17) (dual of [4194356, 4194172, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2177, 4194305, F2, 17) (dual of [4194305, 4194128, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2133, 4194305, F2, 13) (dual of [4194305, 4194172, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 51, F2, 3) (dual of [51, 44, 4]-code or 51-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 C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2184, 4194356, F2, 17) (dual of [4194356, 4194172, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2184, 4194353, F2, 17) (dual of [4194353, 4194169, 18]-code), using
(167, 167+17, 699059)-Net over F2 — Digital
Digital (167, 184, 699059)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2184, 699059, F2, 6, 17) (dual of [(699059, 6), 4194170, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2184, 4194354, F2, 17) (dual of [4194354, 4194170, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2184, 4194356, F2, 17) (dual of [4194356, 4194172, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2177, 4194305, F2, 17) (dual of [4194305, 4194128, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2133, 4194305, F2, 13) (dual of [4194305, 4194172, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 51, F2, 3) (dual of [51, 44, 4]-code or 51-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 C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2184, 4194356, F2, 17) (dual of [4194356, 4194172, 18]-code), using
- OOA 6-folding [i] based on linear OA(2184, 4194354, F2, 17) (dual of [4194354, 4194170, 18]-code), using
(167, 167+17, large)-Net in Base 2 — Upper bound on s
There is no (167, 184, large)-net in base 2, because
- 15 times m-reduction [i] would yield (167, 169, large)-net in base 2, but