Best Known (251−35, 251, s)-Nets in Base 2
(251−35, 251, 966)-Net over F2 — Constructive and digital
Digital (216, 251, 966)-net over F2, using
- 21 times duplication [i] based on digital (215, 250, 966)-net over F2, using
- net defined by OOA [i] based on linear OOA(2250, 966, F2, 35, 35) (dual of [(966, 35), 33560, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2250, 16423, F2, 35) (dual of [16423, 16173, 36]-code), using
- 4 times code embedding in larger space [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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(34) ⊂ Ce(30) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2250, 16423, F2, 35) (dual of [16423, 16173, 36]-code), using
- net defined by OOA [i] based on linear OOA(2250, 966, F2, 35, 35) (dual of [(966, 35), 33560, 36]-NRT-code), using
(251−35, 251, 3581)-Net over F2 — Digital
Digital (216, 251, 3581)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2251, 3581, F2, 4, 35) (dual of [(3581, 4), 14073, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2251, 4108, F2, 4, 35) (dual of [(4108, 4), 16181, 36]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2251, 16432, F2, 35) (dual of [16432, 16181, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2197, 16384, F2, 29) (dual of [16384, 16187, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(34) ⊂ Ce(28) [i] based on
- OOA 4-folding [i] based on linear OA(2251, 16432, F2, 35) (dual of [16432, 16181, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2251, 4108, F2, 4, 35) (dual of [(4108, 4), 16181, 36]-NRT-code), using
(251−35, 251, 191778)-Net in Base 2 — Upper bound on s
There is no (216, 251, 191779)-net in base 2, because
- 1 times m-reduction [i] would yield (216, 250, 191779)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1809 304433 792925 566495 504139 807990 371088 624067 793491 323424 321883 883367 401592 > 2250 [i]