Best Known (238−30, 238, s)-Nets in Base 2
(238−30, 238, 2188)-Net over F2 — Constructive and digital
Digital (208, 238, 2188)-net over F2, using
- net defined by OOA [i] based on linear OOA(2238, 2188, F2, 30, 30) (dual of [(2188, 30), 65402, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2238, 32820, F2, 30) (dual of [32820, 32582, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2238, 32825, F2, 30) (dual of [32825, 32587, 31]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2239, 32826, F2, 31) (dual of [32826, 32587, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2238, 32825, F2, 30) (dual of [32825, 32587, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2238, 32820, F2, 30) (dual of [32820, 32582, 31]-code), using
(238−30, 238, 6651)-Net over F2 — Digital
Digital (208, 238, 6651)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2238, 6651, F2, 4, 30) (dual of [(6651, 4), 26366, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2238, 8206, F2, 4, 30) (dual of [(8206, 4), 32586, 31]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2238, 32824, F2, 30) (dual of [32824, 32586, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2238, 32825, F2, 30) (dual of [32825, 32587, 31]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2239, 32826, F2, 31) (dual of [32826, 32587, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2238, 32825, F2, 30) (dual of [32825, 32587, 31]-code), using
- OOA 4-folding [i] based on linear OA(2238, 32824, F2, 30) (dual of [32824, 32586, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(2238, 8206, F2, 4, 30) (dual of [(8206, 4), 32586, 31]-NRT-code), using
(238−30, 238, 383781)-Net in Base 2 — Upper bound on s
There is no (208, 238, 383782)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 441715 692809 825656 062387 062326 781150 367960 194653 289737 594445 507609 575840 > 2238 [i]