Best Known (248−25, 248, s)-Nets in Base 2
(248−25, 248, 87385)-Net over F2 — Constructive and digital
Digital (223, 248, 87385)-net over F2, using
- net defined by OOA [i] based on linear OOA(2248, 87385, F2, 25, 25) (dual of [(87385, 25), 2184377, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2248, 1048621, F2, 25) (dual of [1048621, 1048373, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2241, 1048577, F2, 25) (dual of [1048577, 1048336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2248, 1048621, F2, 25) (dual of [1048621, 1048373, 26]-code), using
(248−25, 248, 132864)-Net over F2 — Digital
Digital (223, 248, 132864)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 132864, F2, 7, 25) (dual of [(132864, 7), 929800, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2248, 149803, F2, 7, 25) (dual of [(149803, 7), 1048373, 26]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2248, 1048621, F2, 25) (dual of [1048621, 1048373, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2241, 1048577, F2, 25) (dual of [1048577, 1048336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- OOA 7-folding [i] based on linear OA(2248, 1048621, F2, 25) (dual of [1048621, 1048373, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(2248, 149803, F2, 7, 25) (dual of [(149803, 7), 1048373, 26]-NRT-code), using
(248−25, 248, 8309238)-Net in Base 2 — Upper bound on s
There is no (223, 248, 8309239)-net in base 2, because
- 1 times m-reduction [i] would yield (223, 247, 8309239)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 226 156472 631471 260098 563410 492921 162799 426701 115726 113674 926076 309686 699162 > 2247 [i]