Best Known (138, 138+16, s)-Nets in Base 2
(138, 138+16, 65538)-Net over F2 — Constructive and digital
Digital (138, 154, 65538)-net over F2, using
- t-expansion [i] based on digital (137, 154, 65538)-net over F2, using
- net defined by OOA [i] based on linear OOA(2154, 65538, F2, 17, 17) (dual of [(65538, 17), 1113992, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2154, 524305, F2, 17) (dual of [524305, 524151, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2154, 524308, F2, 17) (dual of [524308, 524154, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2153, 524288, F2, 17) (dual of [524288, 524135, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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 X applied to Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2154, 524308, F2, 17) (dual of [524308, 524154, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2154, 524305, F2, 17) (dual of [524305, 524151, 18]-code), using
- net defined by OOA [i] based on linear OOA(2154, 65538, F2, 17, 17) (dual of [(65538, 17), 1113992, 18]-NRT-code), using
(138, 138+16, 104861)-Net over F2 — Digital
Digital (138, 154, 104861)-net over F2, using
- 21 times duplication [i] based on digital (137, 153, 104861)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2153, 104861, F2, 5, 16) (dual of [(104861, 5), 524152, 17]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2153, 524305, F2, 16) (dual of [524305, 524152, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2153, 524308, F2, 16) (dual of [524308, 524155, 17]-code), using
- 1 times truncation [i] based on linear OA(2154, 524309, F2, 17) (dual of [524309, 524155, 18]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2153, 524288, F2, 17) (dual of [524288, 524135, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(220, 21, F2, 19) (dual of [21, 1, 20]-code), using
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- dual of repetition code with length 21 [i]
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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 Ce(16) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2154, 524309, F2, 17) (dual of [524309, 524155, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2153, 524308, F2, 16) (dual of [524308, 524155, 17]-code), using
- OOA 5-folding [i] based on linear OA(2153, 524305, F2, 16) (dual of [524305, 524152, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2153, 104861, F2, 5, 16) (dual of [(104861, 5), 524152, 17]-NRT-code), using
(138, 138+16, 2347012)-Net in Base 2 — Upper bound on s
There is no (138, 154, 2347013)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 22836 022457 615854 515573 058365 831428 323438 566752 > 2154 [i]