Best Known (98, 98+17, s)-Nets in Base 2
(98, 98+17, 2049)-Net over F2 — Constructive and digital
Digital (98, 115, 2049)-net over F2, using
- 21 times duplication [i] based on digital (97, 114, 2049)-net over F2, using
- net defined by OOA [i] based on linear OOA(2114, 2049, F2, 17, 17) (dual of [(2049, 17), 34719, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2114, 16393, F2, 17) (dual of [16393, 16279, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2114, 16399, F2, 17) (dual of [16399, 16285, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2113, 16384, F2, 17) (dual of [16384, 16271, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(299, 16384, F2, 15) (dual of [16384, 16285, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2114, 16399, F2, 17) (dual of [16399, 16285, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2114, 16393, F2, 17) (dual of [16393, 16279, 18]-code), using
- net defined by OOA [i] based on linear OOA(2114, 2049, F2, 17, 17) (dual of [(2049, 17), 34719, 18]-NRT-code), using
(98, 98+17, 3280)-Net over F2 — Digital
Digital (98, 115, 3280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2115, 3280, F2, 5, 17) (dual of [(3280, 5), 16285, 18]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2115, 16400, F2, 17) (dual of [16400, 16285, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2114, 16399, F2, 17) (dual of [16399, 16285, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2113, 16384, F2, 17) (dual of [16384, 16271, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(299, 16384, F2, 15) (dual of [16384, 16285, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(16) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2114, 16399, F2, 17) (dual of [16399, 16285, 18]-code), using
- OOA 5-folding [i] based on linear OA(2115, 16400, F2, 17) (dual of [16400, 16285, 18]-code), using
(98, 98+17, 73332)-Net in Base 2 — Upper bound on s
There is no (98, 115, 73333)-net in base 2, because
- 1 times m-reduction [i] would yield (98, 114, 73333)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 20769 204229 240451 074469 195550 475054 > 2114 [i]