Best Known (242−33, 242, s)-Nets in Base 2
(242−33, 242, 2048)-Net over F2 — Constructive and digital
Digital (209, 242, 2048)-net over F2, using
- 21 times duplication [i] based on digital (208, 241, 2048)-net over F2, using
- net defined by OOA [i] based on linear OOA(2241, 2048, F2, 33, 33) (dual of [(2048, 33), 67343, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2241, 32769, F2, 33) (dual of [32769, 32528, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(2241, 32769, F2, 33) (dual of [32769, 32528, 34]-code), using
- net defined by OOA [i] based on linear OOA(2241, 2048, F2, 33, 33) (dual of [(2048, 33), 67343, 34]-NRT-code), using
(242−33, 242, 5464)-Net over F2 — Digital
Digital (209, 242, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 5464, F2, 6, 33) (dual of [(5464, 6), 32542, 34]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- OOA 6-folding [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
(242−33, 242, 232680)-Net in Base 2 — Upper bound on s
There is no (209, 242, 232681)-net in base 2, because
- 1 times m-reduction [i] would yield (209, 241, 232681)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 533921 359892 225637 661687 662153 695867 321235 036857 890821 727167 849933 419470 > 2241 [i]