Best Known (228−25, 228, s)-Nets in Base 2
(228−25, 228, 21849)-Net over F2 — Constructive and digital
Digital (203, 228, 21849)-net over F2, using
- 23 times duplication [i] based on digital (200, 225, 21849)-net over F2, using
- net defined by OOA [i] based on linear OOA(2225, 21849, F2, 25, 25) (dual of [(21849, 25), 546000, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2225, 262189, F2, 25) (dual of [262189, 261964, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2224, 262188, F2, 25) (dual of [262188, 261964, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2217, 262145, F2, 25) (dual of [262145, 261928, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2181, 262145, F2, 21) (dual of [262145, 261964, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 43, F2, 3) (dual of [43, 36, 4]-code or 43-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
- 1 times code embedding in larger space [i] based on linear OA(2224, 262188, F2, 25) (dual of [262188, 261964, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2225, 262189, F2, 25) (dual of [262189, 261964, 26]-code), using
- net defined by OOA [i] based on linear OOA(2225, 21849, F2, 25, 25) (dual of [(21849, 25), 546000, 26]-NRT-code), using
(228−25, 228, 38953)-Net over F2 — Digital
Digital (203, 228, 38953)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 38953, F2, 6, 25) (dual of [(38953, 6), 233490, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2228, 43698, F2, 6, 25) (dual of [(43698, 6), 261960, 26]-NRT-code), using
- 24 times duplication [i] based on linear OOA(2224, 43698, F2, 6, 25) (dual of [(43698, 6), 261964, 26]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2224, 262188, F2, 25) (dual of [262188, 261964, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2217, 262145, F2, 25) (dual of [262145, 261928, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2181, 262145, F2, 21) (dual of [262145, 261964, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 43, F2, 3) (dual of [43, 36, 4]-code or 43-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 6-folding [i] based on linear OA(2224, 262188, F2, 25) (dual of [262188, 261964, 26]-code), using
- 24 times duplication [i] based on linear OOA(2224, 43698, F2, 6, 25) (dual of [(43698, 6), 261964, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2228, 43698, F2, 6, 25) (dual of [(43698, 6), 261960, 26]-NRT-code), using
(228−25, 228, 2617234)-Net in Base 2 — Upper bound on s
There is no (203, 228, 2617235)-net in base 2, because
- 1 times m-reduction [i] would yield (203, 227, 2617235)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 215 680318 204166 802809 220749 217876 839720 436183 559989 579091 368175 872948 > 2227 [i]