Best Known (113−16, 113, s)-Nets in Base 2
(113−16, 113, 2049)-Net over F2 — Constructive and digital
Digital (97, 113, 2049)-net over F2, using
- net defined by OOA [i] based on linear OOA(2113, 2049, F2, 16, 16) (dual of [(2049, 16), 32671, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2113, 16392, F2, 16) (dual of [16392, 16279, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 16398, F2, 16) (dual of [16398, 16285, 17]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(2114, 16399, F2, 17) (dual of [16399, 16285, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 16398, F2, 16) (dual of [16398, 16285, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2113, 16392, F2, 16) (dual of [16392, 16279, 17]-code), using
(113−16, 113, 4099)-Net over F2 — Digital
Digital (97, 113, 4099)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2113, 4099, F2, 4, 16) (dual of [(4099, 4), 16283, 17]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2113, 16396, F2, 16) (dual of [16396, 16283, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 16398, F2, 16) (dual of [16398, 16285, 17]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(2114, 16399, F2, 17) (dual of [16399, 16285, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 16398, F2, 16) (dual of [16398, 16285, 17]-code), using
- OOA 4-folding [i] based on linear OA(2113, 16396, F2, 16) (dual of [16396, 16283, 17]-code), using
(113−16, 113, 67245)-Net in Base 2 — Upper bound on s
There is no (97, 113, 67246)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 10384 968748 749320 341223 721142 757083 > 2113 [i]