Best Known (237−33, 237, s)-Nets in Base 2
(237−33, 237, 1026)-Net over F2 — Constructive and digital
Digital (204, 237, 1026)-net over F2, using
- 26 times duplication [i] based on digital (198, 231, 1026)-net over F2, using
- net defined by OOA [i] based on linear OOA(2231, 1026, F2, 33, 33) (dual of [(1026, 33), 33627, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2231, 16417, F2, 33) (dual of [16417, 16186, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2197, 16385, F2, 29) (dual of [16385, 16188, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- OOA 16-folding and stacking with additional row [i] based on linear OA(2231, 16417, F2, 33) (dual of [16417, 16186, 34]-code), using
- net defined by OOA [i] based on linear OOA(2231, 1026, F2, 33, 33) (dual of [(1026, 33), 33627, 34]-NRT-code), using
(237−33, 237, 3573)-Net over F2 — Digital
Digital (204, 237, 3573)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2237, 3573, F2, 4, 33) (dual of [(3573, 4), 14055, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2237, 4108, F2, 4, 33) (dual of [(4108, 4), 16195, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2237, 16432, F2, 33) (dual of [16432, 16195, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- linear OA(2225, 16384, F2, 33) (dual of [16384, 16159, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- OOA 4-folding [i] based on linear OA(2237, 16432, F2, 33) (dual of [16432, 16195, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2237, 4108, F2, 4, 33) (dual of [(4108, 4), 16195, 34]-NRT-code), using
(237−33, 237, 187359)-Net in Base 2 — Upper bound on s
There is no (204, 237, 187360)-net in base 2, because
- 1 times m-reduction [i] would yield (204, 236, 187360)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 110428 685261 676907 541021 821006 453989 892684 942973 021390 329227 386173 052335 > 2236 [i]