Best Known (237−30, 237, s)-Nets in Base 2
(237−30, 237, 2187)-Net over F2 — Constructive and digital
Digital (207, 237, 2187)-net over F2, using
- 23 times duplication [i] based on digital (204, 234, 2187)-net over F2, using
- t-expansion [i] based on digital (203, 234, 2187)-net over F2, using
- net defined by OOA [i] based on linear OOA(2234, 2187, F2, 31, 31) (dual of [(2187, 31), 67563, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2234, 32806, F2, 31) (dual of [32806, 32572, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- 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(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(27, 37, F2, 3) (dual of [37, 30, 4]-code or 37-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(30) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2234, 32806, F2, 31) (dual of [32806, 32572, 32]-code), using
- net defined by OOA [i] based on linear OOA(2234, 2187, F2, 31, 31) (dual of [(2187, 31), 67563, 32]-NRT-code), using
- t-expansion [i] based on digital (203, 234, 2187)-net over F2, using
(237−30, 237, 6563)-Net over F2 — Digital
Digital (207, 237, 6563)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2237, 6563, F2, 5, 30) (dual of [(6563, 5), 32578, 31]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2237, 32815, F2, 30) (dual of [32815, 32578, 31]-code), using
- 1 times truncation [i] based on linear OA(2238, 32816, F2, 31) (dual of [32816, 32578, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- 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(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(24) [i] based on
- 1 times truncation [i] based on linear OA(2238, 32816, F2, 31) (dual of [32816, 32578, 32]-code), using
- OOA 5-folding [i] based on linear OA(2237, 32815, F2, 30) (dual of [32815, 32578, 31]-code), using
(237−30, 237, 366449)-Net in Base 2 — Upper bound on s
There is no (207, 237, 366450)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 220857 602327 355813 492404 707918 871607 923967 356345 875356 039530 110561 918936 > 2237 [i]