Best Known (239−30, 239, s)-Nets in Base 2
(239−30, 239, 2188)-Net over F2 — Constructive and digital
Digital (209, 239, 2188)-net over F2, using
- t-expansion [i] based on digital (208, 239, 2188)-net over F2, using
- net defined by OOA [i] based on linear OOA(2239, 2188, F2, 31, 31) (dual of [(2188, 31), 67589, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2239, 32821, F2, 31) (dual of [32821, 32582, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2239, 32826, F2, 31) (dual of [32826, 32587, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(213, 58, F2, 5) (dual of [58, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2239, 32826, F2, 31) (dual of [32826, 32587, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2239, 32821, F2, 31) (dual of [32821, 32582, 32]-code), using
- net defined by OOA [i] based on linear OOA(2239, 2188, F2, 31, 31) (dual of [(2188, 31), 67589, 32]-NRT-code), using
(239−30, 239, 6839)-Net over F2 — Digital
Digital (209, 239, 6839)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2239, 6839, F2, 4, 30) (dual of [(6839, 4), 27117, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2239, 8206, F2, 4, 30) (dual of [(8206, 4), 32585, 31]-NRT-code), using
- strength reduction [i] based on linear OOA(2239, 8206, F2, 4, 31) (dual of [(8206, 4), 32585, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2239, 32824, F2, 31) (dual of [32824, 32585, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2239, 32826, F2, 31) (dual of [32826, 32587, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(213, 58, F2, 5) (dual of [58, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2239, 32826, F2, 31) (dual of [32826, 32587, 32]-code), using
- OOA 4-folding [i] based on linear OA(2239, 32824, F2, 31) (dual of [32824, 32585, 32]-code), using
- strength reduction [i] based on linear OOA(2239, 8206, F2, 4, 31) (dual of [(8206, 4), 32585, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2239, 8206, F2, 4, 30) (dual of [(8206, 4), 32585, 31]-NRT-code), using
(239−30, 239, 401933)-Net in Base 2 — Upper bound on s
There is no (209, 239, 401934)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 883442 150812 942515 822834 697412 675395 170852 436747 638056 036210 204675 657820 > 2239 [i]