Best Known (153−17, 153, s)-Nets in Base 2
(153−17, 153, 65536)-Net over F2 — Constructive and digital
Digital (136, 153, 65536)-net over F2, using
- net defined by OOA [i] based on linear OOA(2153, 65536, F2, 17, 17) (dual of [(65536, 17), 1113959, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2153, 524289, F2, 17) (dual of [524289, 524136, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(2153, 524289, F2, 17) (dual of [524289, 524136, 18]-code), using
(153−17, 153, 87381)-Net over F2 — Digital
Digital (136, 153, 87381)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2153, 87381, F2, 6, 17) (dual of [(87381, 6), 524133, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2153, 524286, F2, 17) (dual of [524286, 524133, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2153, 524288, F2, 17) (dual of [524288, 524135, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(2153, 524288, F2, 17) (dual of [524288, 524135, 18]-code), using
- OOA 6-folding [i] based on linear OA(2153, 524286, F2, 17) (dual of [524286, 524133, 18]-code), using
(153−17, 153, 1973592)-Net in Base 2 — Upper bound on s
There is no (136, 153, 1973593)-net in base 2, because
- 1 times m-reduction [i] would yield (136, 152, 1973593)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5709 005878 879544 471246 580360 139945 234985 337895 > 2152 [i]