Best Known (168−17, 168, s)-Nets in Base 2
(168−17, 168, 131077)-Net over F2 — Constructive and digital
Digital (151, 168, 131077)-net over F2, using
- net defined by OOA [i] based on linear OOA(2168, 131077, F2, 17, 17) (dual of [(131077, 17), 2228141, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2168, 1048617, F2, 17) (dual of [1048617, 1048449, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 1048624, F2, 17) (dual of [1048624, 1048456, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2161, 1048577, F2, 17) (dual of [1048577, 1048416, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2121, 1048577, F2, 13) (dual of [1048577, 1048456, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-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(2168, 1048624, F2, 17) (dual of [1048624, 1048456, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2168, 1048617, F2, 17) (dual of [1048617, 1048449, 18]-code), using
(168−17, 168, 174770)-Net over F2 — Digital
Digital (151, 168, 174770)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2168, 174770, F2, 6, 17) (dual of [(174770, 6), 1048452, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2168, 1048620, F2, 17) (dual of [1048620, 1048452, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 1048624, F2, 17) (dual of [1048624, 1048456, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2161, 1048577, F2, 17) (dual of [1048577, 1048416, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2121, 1048577, F2, 13) (dual of [1048577, 1048456, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-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(2168, 1048624, F2, 17) (dual of [1048624, 1048456, 18]-code), using
- OOA 6-folding [i] based on linear OA(2168, 1048620, F2, 17) (dual of [1048620, 1048452, 18]-code), using
(168−17, 168, 7239199)-Net in Base 2 — Upper bound on s
There is no (151, 168, 7239200)-net in base 2, because
- 1 times m-reduction [i] would yield (151, 167, 7239200)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 187 072348 098626 238369 740048 779585 574445 983607 397141 > 2167 [i]