Best Known (158, 158+19, s)-Nets in Base 2
(158, 158+19, 58256)-Net over F2 — Constructive and digital
Digital (158, 177, 58256)-net over F2, using
- 24 times duplication [i] based on digital (154, 173, 58256)-net over F2, using
- net defined by OOA [i] based on linear OOA(2173, 58256, F2, 19, 19) (dual of [(58256, 19), 1106691, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2173, 524305, F2, 19) (dual of [524305, 524132, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2173, 524308, F2, 19) (dual of [524308, 524135, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2173, 524308, F2, 19) (dual of [524308, 524135, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2173, 524305, F2, 19) (dual of [524305, 524132, 20]-code), using
- net defined by OOA [i] based on linear OOA(2173, 58256, F2, 19, 19) (dual of [(58256, 19), 1106691, 20]-NRT-code), using
(158, 158+19, 87385)-Net over F2 — Digital
Digital (158, 177, 87385)-net over F2, using
- 22 times duplication [i] based on digital (156, 175, 87385)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2175, 87385, F2, 6, 19) (dual of [(87385, 6), 524135, 20]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2174, 87385, F2, 6, 19) (dual of [(87385, 6), 524136, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2174, 524310, F2, 19) (dual of [524310, 524136, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2173, 524309, F2, 19) (dual of [524309, 524136, 20]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2173, 524309, F2, 19) (dual of [524309, 524136, 20]-code), using
- OOA 6-folding [i] based on linear OA(2174, 524310, F2, 19) (dual of [524310, 524136, 20]-code), using
- 21 times duplication [i] based on linear OOA(2174, 87385, F2, 6, 19) (dual of [(87385, 6), 524136, 20]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2175, 87385, F2, 6, 19) (dual of [(87385, 6), 524135, 20]-NRT-code), using
(158, 158+19, 3195644)-Net in Base 2 — Upper bound on s
There is no (158, 177, 3195645)-net in base 2, because
- 1 times m-reduction [i] would yield (158, 176, 3195645)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 95781 055949 182010 743929 189073 273354 090833 026583 714110 > 2176 [i]