Best Known (157−16, 157, s)-Nets in Base 2
(157−16, 157, 65539)-Net over F2 — Constructive and digital
Digital (141, 157, 65539)-net over F2, using
- net defined by OOA [i] based on linear OOA(2157, 65539, F2, 16, 16) (dual of [(65539, 16), 1048467, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2157, 524312, F2, 16) (dual of [524312, 524155, 17]-code), using
- 4 times code embedding in larger space [i] 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
- 4 times code embedding in larger space [i] based on linear OA(2153, 524308, F2, 16) (dual of [524308, 524155, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2157, 524312, F2, 16) (dual of [524312, 524155, 17]-code), using
(157−16, 157, 104862)-Net over F2 — Digital
Digital (141, 157, 104862)-net over F2, using
- 22 times duplication [i] based on digital (139, 155, 104862)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 104862, F2, 5, 16) (dual of [(104862, 5), 524155, 17]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2155, 524310, F2, 16) (dual of [524310, 524155, 17]-code), using
- 2 times code embedding in larger space [i] 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
- 2 times code embedding in larger space [i] based on linear OA(2153, 524308, F2, 16) (dual of [524308, 524155, 17]-code), using
- OOA 5-folding [i] based on linear OA(2155, 524310, F2, 16) (dual of [524310, 524155, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 104862, F2, 5, 16) (dual of [(104862, 5), 524155, 17]-NRT-code), using
(157−16, 157, 3043701)-Net in Base 2 — Upper bound on s
There is no (141, 157, 3043702)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 182687 840025 252314 174424 056053 057238 098552 184830 > 2157 [i]