Best Known (177−25, 177, s)-Nets in Base 2
(177−25, 177, 1368)-Net over F2 — Constructive and digital
Digital (152, 177, 1368)-net over F2, using
- 22 times duplication [i] based on digital (150, 175, 1368)-net over F2, using
- net defined by OOA [i] based on linear OOA(2175, 1368, F2, 25, 25) (dual of [(1368, 25), 34025, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2175, 16417, F2, 25) (dual of [16417, 16242, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 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(2141, 16385, F2, 21) (dual of [16385, 16244, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(2175, 16417, F2, 25) (dual of [16417, 16242, 26]-code), using
- net defined by OOA [i] based on linear OOA(2175, 1368, F2, 25, 25) (dual of [(1368, 25), 34025, 26]-NRT-code), using
(177−25, 177, 3307)-Net over F2 — Digital
Digital (152, 177, 3307)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2177, 3307, F2, 4, 25) (dual of [(3307, 4), 13051, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2177, 4105, F2, 4, 25) (dual of [(4105, 4), 16243, 26]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2176, 4105, F2, 4, 25) (dual of [(4105, 4), 16244, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2176, 16420, F2, 25) (dual of [16420, 16244, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 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(2141, 16385, F2, 21) (dual of [16385, 16244, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
- OOA 4-folding [i] based on linear OA(2176, 16420, F2, 25) (dual of [16420, 16244, 26]-code), using
- 21 times duplication [i] based on linear OOA(2176, 4105, F2, 4, 25) (dual of [(4105, 4), 16244, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2177, 4105, F2, 4, 25) (dual of [(4105, 4), 16243, 26]-NRT-code), using
(177−25, 177, 137534)-Net in Base 2 — Upper bound on s
There is no (152, 177, 137535)-net in base 2, because
- 1 times m-reduction [i] would yield (152, 176, 137535)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 95782 210244 342842 762991 077794 021008 975484 392022 428493 > 2176 [i]