Best Known (179, 179+28, s)-Nets in Base 2
(179, 179+28, 1173)-Net over F2 — Constructive and digital
Digital (179, 207, 1173)-net over F2, using
- t-expansion [i] based on digital (178, 207, 1173)-net over F2, using
- net defined by OOA [i] based on linear OOA(2207, 1173, F2, 29, 29) (dual of [(1173, 29), 33810, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2207, 16423, F2, 29) (dual of [16423, 16216, 30]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2204, 16420, F2, 29) (dual of [16420, 16216, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2197, 16385, F2, 29) (dual of [16385, 16188, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2169, 16385, F2, 25) (dual of [16385, 16216, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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([0,14]) ⊂ C([0,12]) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2204, 16420, F2, 29) (dual of [16420, 16216, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2207, 16423, F2, 29) (dual of [16423, 16216, 30]-code), using
- net defined by OOA [i] based on linear OOA(2207, 1173, F2, 29, 29) (dual of [(1173, 29), 33810, 30]-NRT-code), using
(179, 179+28, 4105)-Net over F2 — Digital
Digital (179, 207, 4105)-net over F2, using
- 21 times duplication [i] based on digital (178, 206, 4105)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 4105, F2, 4, 28) (dual of [(4105, 4), 16214, 29]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2204, 4105, F2, 4, 28) (dual of [(4105, 4), 16216, 29]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2204, 16420, F2, 28) (dual of [16420, 16216, 29]-code), using
- strength reduction [i] based on linear OA(2204, 16420, F2, 29) (dual of [16420, 16216, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2197, 16385, F2, 29) (dual of [16385, 16188, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2169, 16385, F2, 25) (dual of [16385, 16216, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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([0,14]) ⊂ C([0,12]) [i] based on
- strength reduction [i] based on linear OA(2204, 16420, F2, 29) (dual of [16420, 16216, 30]-code), using
- OOA 4-folding [i] based on linear OA(2204, 16420, F2, 28) (dual of [16420, 16216, 29]-code), using
- 22 times duplication [i] based on linear OOA(2204, 4105, F2, 4, 28) (dual of [(4105, 4), 16216, 29]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 4105, F2, 4, 28) (dual of [(4105, 4), 16214, 29]-NRT-code), using
(179, 179+28, 170745)-Net in Base 2 — Upper bound on s
There is no (179, 207, 170746)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 205 697536 436724 519065 909258 706088 225917 856158 844513 043206 982536 > 2207 [i]