Best Known (140, 140+19, s)-Nets in Base 2
(140, 140+19, 14566)-Net over F2 — Constructive and digital
Digital (140, 159, 14566)-net over F2, using
- net defined by OOA [i] based on linear OOA(2159, 14566, F2, 19, 19) (dual of [(14566, 19), 276595, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2159, 131095, F2, 19) (dual of [131095, 130936, 20]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2156, 131092, F2, 19) (dual of [131092, 130936, 20]-code), using
- adding a parity check bit [i] based on linear OA(2155, 131091, F2, 18) (dual of [131091, 130936, 19]-code), using
- construction X4 applied to C([0,18]) ⊂ C([1,16]) [i] based on
- linear OA(2154, 131071, F2, 19) (dual of [131071, 130917, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2136, 131071, F2, 16) (dual of [131071, 130935, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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([0,18]) ⊂ C([1,16]) [i] based on
- adding a parity check bit [i] based on linear OA(2155, 131091, F2, 18) (dual of [131091, 130936, 19]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2156, 131092, F2, 19) (dual of [131092, 130936, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2159, 131095, F2, 19) (dual of [131095, 130936, 20]-code), using
(140, 140+19, 21849)-Net over F2 — Digital
Digital (140, 159, 21849)-net over F2, using
- 21 times duplication [i] based on digital (139, 158, 21849)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2158, 21849, F2, 6, 19) (dual of [(21849, 6), 130936, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2158, 131094, F2, 19) (dual of [131094, 130936, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2156, 131092, F2, 19) (dual of [131092, 130936, 20]-code), using
- adding a parity check bit [i] based on linear OA(2155, 131091, F2, 18) (dual of [131091, 130936, 19]-code), using
- construction X4 applied to C([0,18]) ⊂ C([1,16]) [i] based on
- linear OA(2154, 131071, F2, 19) (dual of [131071, 130917, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2136, 131071, F2, 16) (dual of [131071, 130935, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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([0,18]) ⊂ C([1,16]) [i] based on
- adding a parity check bit [i] based on linear OA(2155, 131091, F2, 18) (dual of [131091, 130936, 19]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2156, 131092, F2, 19) (dual of [131092, 130936, 20]-code), using
- OOA 6-folding [i] based on linear OA(2158, 131094, F2, 19) (dual of [131094, 130936, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2158, 21849, F2, 6, 19) (dual of [(21849, 6), 130936, 20]-NRT-code), using
(140, 140+19, 798901)-Net in Base 2 — Upper bound on s
There is no (140, 159, 798902)-net in base 2, because
- 1 times m-reduction [i] would yield (140, 158, 798902)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 365377 789900 293226 858017 683078 422380 564447 573706 > 2158 [i]