Best Known (48−11, 48, s)-Nets in Base 2
(48−11, 48, 106)-Net over F2 — Constructive and digital
Digital (37, 48, 106)-net over F2, using
- net defined by OOA [i] based on linear OOA(248, 106, F2, 11, 11) (dual of [(106, 11), 1118, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(248, 531, F2, 11) (dual of [531, 483, 12]-code), using
- construction XX applied to C1 = C([509,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([509,8]) [i] based on
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(246, 511, F2, 11) (dual of [511, 465, 12]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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
- linear OA(21, 10, F2, 1) (dual of [10, 9, 2]-code) (see above)
- construction XX applied to C1 = C([509,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([509,8]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(248, 531, F2, 11) (dual of [531, 483, 12]-code), using
(48−11, 48, 192)-Net over F2 — Digital
Digital (37, 48, 192)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(248, 192, F2, 2, 11) (dual of [(192, 2), 336, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(248, 265, F2, 2, 11) (dual of [(265, 2), 482, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(248, 530, F2, 11) (dual of [530, 482, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(248, 531, F2, 11) (dual of [531, 483, 12]-code), using
- construction XX applied to C1 = C([509,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([509,8]) [i] based on
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(246, 511, F2, 11) (dual of [511, 465, 12]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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
- linear OA(21, 10, F2, 1) (dual of [10, 9, 2]-code) (see above)
- construction XX applied to C1 = C([509,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([509,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(248, 531, F2, 11) (dual of [531, 483, 12]-code), using
- OOA 2-folding [i] based on linear OA(248, 530, F2, 11) (dual of [530, 482, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(248, 265, F2, 2, 11) (dual of [(265, 2), 482, 12]-NRT-code), using
(48−11, 48, 1753)-Net in Base 2 — Upper bound on s
There is no (37, 48, 1754)-net in base 2, because
- 1 times m-reduction [i] would yield (37, 47, 1754)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 141 123991 953902 > 247 [i]