Best Known (121−16, 121, s)-Nets in Base 2
(121−16, 121, 4097)-Net over F2 — Constructive and digital
Digital (105, 121, 4097)-net over F2, using
- net defined by OOA [i] based on linear OOA(2121, 4097, F2, 16, 16) (dual of [(4097, 16), 65431, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2121, 32776, F2, 16) (dual of [32776, 32655, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2121, 32783, F2, 16) (dual of [32783, 32662, 17]-code), using
- 1 times truncation [i] based on linear OA(2122, 32784, F2, 17) (dual of [32784, 32662, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2121, 32768, F2, 17) (dual of [32768, 32647, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(2122, 32784, F2, 17) (dual of [32784, 32662, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2121, 32783, F2, 16) (dual of [32783, 32662, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2121, 32776, F2, 16) (dual of [32776, 32655, 17]-code), using
(121−16, 121, 7799)-Net over F2 — Digital
Digital (105, 121, 7799)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2121, 7799, F2, 4, 16) (dual of [(7799, 4), 31075, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2121, 8195, F2, 4, 16) (dual of [(8195, 4), 32659, 17]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2121, 32780, F2, 16) (dual of [32780, 32659, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2121, 32783, F2, 16) (dual of [32783, 32662, 17]-code), using
- 1 times truncation [i] based on linear OA(2122, 32784, F2, 17) (dual of [32784, 32662, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2121, 32768, F2, 17) (dual of [32768, 32647, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(2122, 32784, F2, 17) (dual of [32784, 32662, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2121, 32783, F2, 16) (dual of [32783, 32662, 17]-code), using
- OOA 4-folding [i] based on linear OA(2121, 32780, F2, 16) (dual of [32780, 32659, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(2121, 8195, F2, 4, 16) (dual of [(8195, 4), 32659, 17]-NRT-code), using
(121−16, 121, 134502)-Net in Base 2 — Upper bound on s
There is no (105, 121, 134503)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 658473 151481 574766 254954 283472 212796 > 2121 [i]