Best Known (178, 207, s)-Nets in Base 2
(178, 207, 1173)-Net over F2 — Constructive and digital
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
(178, 207, 3413)-Net over F2 — Digital
Digital (178, 207, 3413)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2207, 3413, F2, 4, 29) (dual of [(3413, 4), 13445, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2207, 4105, F2, 4, 29) (dual of [(4105, 4), 16213, 30]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2204, 4105, F2, 4, 29) (dual of [(4105, 4), 16216, 30]-NRT-code), using
- OOA 4-folding [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
- OOA 4-folding [i] based on linear OA(2204, 16420, F2, 29) (dual of [16420, 16216, 30]-code), using
- 23 times duplication [i] based on linear OOA(2204, 4105, F2, 4, 29) (dual of [(4105, 4), 16216, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2207, 4105, F2, 4, 29) (dual of [(4105, 4), 16213, 30]-NRT-code), using
(178, 207, 162496)-Net in Base 2 — Upper bound on s
There is no (178, 207, 162497)-net in base 2, because
- 1 times m-reduction [i] would yield (178, 206, 162497)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 102 847435 200389 635458 958516 308097 556603 816936 189899 539761 662368 > 2206 [i]