Best Known (250−35, 250, s)-Nets in Base 2
(250−35, 250, 966)-Net over F2 — Constructive and digital
Digital (215, 250, 966)-net over F2, using
- net defined by OOA [i] based on linear OOA(2250, 966, F2, 35, 35) (dual of [(966, 35), 33560, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2250, 16423, F2, 35) (dual of [16423, 16173, 36]-code), using
- 4 times code embedding in larger space [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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 Ce(34) ⊂ Ce(30) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2250, 16423, F2, 35) (dual of [16423, 16173, 36]-code), using
(250−35, 250, 3498)-Net over F2 — Digital
Digital (215, 250, 3498)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, 3498, F2, 4, 35) (dual of [(3498, 4), 13742, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2250, 4105, F2, 4, 35) (dual of [(4105, 4), 16170, 36]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2247, 4105, F2, 4, 35) (dual of [(4105, 4), 16173, 36]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2247, 16420, F2, 35) (dual of [16420, 16173, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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 Ce(34) ⊂ Ce(30) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- OOA 4-folding [i] based on linear OA(2247, 16420, F2, 35) (dual of [16420, 16173, 36]-code), using
- 23 times duplication [i] based on linear OOA(2247, 4105, F2, 4, 35) (dual of [(4105, 4), 16173, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2250, 4105, F2, 4, 35) (dual of [(4105, 4), 16170, 36]-NRT-code), using
(250−35, 250, 184115)-Net in Base 2 — Upper bound on s
There is no (215, 250, 184116)-net in base 2, because
- 1 times m-reduction [i] would yield (215, 249, 184116)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 904 669351 098151 466020 607057 554198 542989 053849 051556 313327 151597 519432 985027 > 2249 [i]