Best Known (172−18, 172, s)-Nets in Base 2
(172−18, 172, 58256)-Net over F2 — Constructive and digital
Digital (154, 172, 58256)-net over F2, using
- net defined by OOA [i] based on linear OOA(2172, 58256, F2, 18, 18) (dual of [(58256, 18), 1048436, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2172, 524304, F2, 18) (dual of [524304, 524132, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 524308, F2, 18) (dual of [524308, 524136, 19]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(2173, 524309, F2, 19) (dual of [524309, 524136, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 524308, F2, 18) (dual of [524308, 524136, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2172, 524304, F2, 18) (dual of [524304, 524132, 19]-code), using
(172−18, 172, 87384)-Net over F2 — Digital
Digital (154, 172, 87384)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 87384, F2, 6, 18) (dual of [(87384, 6), 524132, 19]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2172, 524304, F2, 18) (dual of [524304, 524132, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 524308, F2, 18) (dual of [524308, 524136, 19]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(2173, 524309, F2, 19) (dual of [524309, 524136, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 524308, F2, 18) (dual of [524308, 524136, 19]-code), using
- OOA 6-folding [i] based on linear OA(2172, 524304, F2, 18) (dual of [524304, 524132, 19]-code), using
(172−18, 172, 2348371)-Net in Base 2 — Upper bound on s
There is no (154, 172, 2348372)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5986 329898 069823 799841 783361 170463 268308 679119 144398 > 2172 [i]