Best Known (109, 109+17, s)-Nets in Base 2
(109, 109+17, 4098)-Net over F2 — Constructive and digital
Digital (109, 126, 4098)-net over F2, using
- 23 times duplication [i] based on digital (106, 123, 4098)-net over F2, using
- net defined by OOA [i] based on linear OOA(2123, 4098, F2, 17, 17) (dual of [(4098, 17), 69543, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2123, 32785, F2, 17) (dual of [32785, 32662, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2123, 32786, F2, 17) (dual of [32786, 32663, 18]-code), using
- adding a parity check bit [i] based on linear OA(2122, 32785, F2, 16) (dual of [32785, 32663, 17]-code), using
- construction X4 applied to C([0,16]) ⊂ C([1,14]) [i] based on
- linear OA(2121, 32767, F2, 17) (dual of [32767, 32646, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2105, 32767, F2, 14) (dual of [32767, 32662, 15]-code), using 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(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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 X4 applied to C([0,16]) ⊂ C([1,14]) [i] based on
- adding a parity check bit [i] based on linear OA(2122, 32785, F2, 16) (dual of [32785, 32663, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2123, 32786, F2, 17) (dual of [32786, 32663, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2123, 32785, F2, 17) (dual of [32785, 32662, 18]-code), using
- net defined by OOA [i] based on linear OOA(2123, 4098, F2, 17, 17) (dual of [(4098, 17), 69543, 18]-NRT-code), using
(109, 109+17, 6557)-Net over F2 — Digital
Digital (109, 126, 6557)-net over F2, using
- 23 times duplication [i] based on digital (106, 123, 6557)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2123, 6557, F2, 5, 17) (dual of [(6557, 5), 32662, 18]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2123, 32785, F2, 17) (dual of [32785, 32662, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2123, 32786, F2, 17) (dual of [32786, 32663, 18]-code), using
- adding a parity check bit [i] based on linear OA(2122, 32785, F2, 16) (dual of [32785, 32663, 17]-code), using
- construction X4 applied to C([0,16]) ⊂ C([1,14]) [i] based on
- linear OA(2121, 32767, F2, 17) (dual of [32767, 32646, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2105, 32767, F2, 14) (dual of [32767, 32662, 15]-code), using 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(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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 X4 applied to C([0,16]) ⊂ C([1,14]) [i] based on
- adding a parity check bit [i] based on linear OA(2122, 32785, F2, 16) (dual of [32785, 32663, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2123, 32786, F2, 17) (dual of [32786, 32663, 18]-code), using
- OOA 5-folding [i] based on linear OA(2123, 32785, F2, 17) (dual of [32785, 32662, 18]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2123, 6557, F2, 5, 17) (dual of [(6557, 5), 32662, 18]-NRT-code), using
(109, 109+17, 190220)-Net in Base 2 — Upper bound on s
There is no (109, 126, 190221)-net in base 2, because
- 1 times m-reduction [i] would yield (109, 125, 190221)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 42 536053 527433 737476 640481 257639 914543 > 2125 [i]