Best Known (241−35, 241, s)-Nets in Base 2
(241−35, 241, 964)-Net over F2 — Constructive and digital
Digital (206, 241, 964)-net over F2, using
- 21 times duplication [i] based on digital (205, 240, 964)-net over F2, using
- net defined by OOA [i] based on linear OOA(2240, 964, F2, 35, 35) (dual of [(964, 35), 33500, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2240, 16389, F2, 35) (dual of [16389, 16149, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2240, 16399, F2, 35) (dual of [16399, 16159, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [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(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(21, 15, F2, 1) (dual of [15, 14, 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(34) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(2240, 16399, F2, 35) (dual of [16399, 16159, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2240, 16389, F2, 35) (dual of [16389, 16149, 36]-code), using
- net defined by OOA [i] based on linear OOA(2240, 964, F2, 35, 35) (dual of [(964, 35), 33500, 36]-NRT-code), using
(241−35, 241, 3245)-Net over F2 — Digital
Digital (206, 241, 3245)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 3245, F2, 5, 35) (dual of [(3245, 5), 15984, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2241, 3280, F2, 5, 35) (dual of [(3280, 5), 16159, 36]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2241, 16400, F2, 35) (dual of [16400, 16159, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2240, 16399, F2, 35) (dual of [16399, 16159, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [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(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(21, 15, F2, 1) (dual of [15, 14, 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(34) ⊂ Ce(32) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2240, 16399, F2, 35) (dual of [16399, 16159, 36]-code), using
- OOA 5-folding [i] based on linear OA(2241, 16400, F2, 35) (dual of [16400, 16159, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2241, 3280, F2, 5, 35) (dual of [(3280, 5), 16159, 36]-NRT-code), using
(241−35, 241, 127554)-Net in Base 2 — Upper bound on s
There is no (206, 241, 127555)-net in base 2, because
- 1 times m-reduction [i] would yield (206, 240, 127555)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 766995 806733 069941 720456 466097 157107 336372 028591 005470 898213 753850 431408 > 2240 [i]