Best Known (177, 206, s)-Nets in Base 2
(177, 206, 1172)-Net over F2 — Constructive and digital
Digital (177, 206, 1172)-net over F2, using
- 23 times duplication [i] based on digital (174, 203, 1172)-net over F2, using
- net defined by OOA [i] based on linear OOA(2203, 1172, F2, 29, 29) (dual of [(1172, 29), 33785, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2203, 16409, F2, 29) (dual of [16409, 16206, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2203, 16417, F2, 29) (dual of [16417, 16214, 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2203, 16417, F2, 29) (dual of [16417, 16214, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2203, 16409, F2, 29) (dual of [16409, 16206, 30]-code), using
- net defined by OOA [i] based on linear OOA(2203, 1172, F2, 29, 29) (dual of [(1172, 29), 33785, 30]-NRT-code), using
(177, 206, 3315)-Net over F2 — Digital
Digital (177, 206, 3315)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 3315, F2, 4, 29) (dual of [(3315, 4), 13054, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 4105, F2, 4, 29) (dual of [(4105, 4), 16214, 30]-NRT-code), using
- 22 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
- 22 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(2206, 4105, F2, 4, 29) (dual of [(4105, 4), 16214, 30]-NRT-code), using
(177, 206, 154646)-Net in Base 2 — Upper bound on s
There is no (177, 206, 154647)-net in base 2, because
- 1 times m-reduction [i] would yield (177, 205, 154647)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 51 425499 710226 986459 310465 859532 910979 600376 322614 600906 601618 > 2205 [i]