Best Known (49−11, 49, s)-Nets in Base 2
(49−11, 49, 106)-Net over F2 — Constructive and digital
Digital (38, 49, 106)-net over F2, using
- 21 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(248, 106, F2, 11, 11) (dual of [(106, 11), 1118, 12]-NRT-code), using
(49−11, 49, 210)-Net over F2 — Digital
Digital (38, 49, 210)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(249, 210, F2, 2, 11) (dual of [(210, 2), 371, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(249, 266, F2, 2, 11) (dual of [(266, 2), 483, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(249, 532, F2, 11) (dual of [532, 483, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(249, 533, F2, 11) (dual of [533, 484, 12]-code), using
- adding a parity check bit [i] based on linear OA(248, 532, F2, 10) (dual of [532, 484, 11]-code), using
- construction XX applied to C1 = C([509,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), 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(236, 511, F2, 8) (dual of [511, 475, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(227, 511, F2, 6) (dual of [511, 484, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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), 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 (see above)
- construction XX applied to C1 = C([509,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([509,8]) [i] based on
- adding a parity check bit [i] based on linear OA(248, 532, F2, 10) (dual of [532, 484, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(249, 533, F2, 11) (dual of [533, 484, 12]-code), using
- OOA 2-folding [i] based on linear OA(249, 532, F2, 11) (dual of [532, 483, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(249, 266, F2, 2, 11) (dual of [(266, 2), 483, 12]-NRT-code), using
(49−11, 49, 2014)-Net in Base 2 — Upper bound on s
There is no (38, 49, 2015)-net in base 2, because
- 1 times m-reduction [i] would yield (38, 48, 2015)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 281 652345 585404 > 248 [i]