Best Known (55−11, 55, s)-Nets in Base 2
(55−11, 55, 209)-Net over F2 — Constructive and digital
Digital (44, 55, 209)-net over F2, using
- 21 times duplication [i] based on digital (43, 54, 209)-net over F2, using
- net defined by OOA [i] based on linear OOA(254, 209, F2, 11, 11) (dual of [(209, 11), 2245, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(254, 1046, F2, 11) (dual of [1046, 992, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(254, 1047, F2, 11) (dual of [1047, 993, 12]-code), using
- adding a parity check bit [i] based on linear OA(253, 1046, F2, 10) (dual of [1046, 993, 11]-code), using
- construction XX applied to C1 = C([1021,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(240, 1023, F2, 8) (dual of [1023, 983, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(251, 1023, F2, 11) (dual of [1023, 972, 12]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(230, 1023, F2, 6) (dual of [1023, 993, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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, 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 (see above)
- construction XX applied to C1 = C([1021,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- adding a parity check bit [i] based on linear OA(253, 1046, F2, 10) (dual of [1046, 993, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(254, 1047, F2, 11) (dual of [1047, 993, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(254, 1046, F2, 11) (dual of [1046, 992, 12]-code), using
- net defined by OOA [i] based on linear OOA(254, 209, F2, 11, 11) (dual of [(209, 11), 2245, 12]-NRT-code), using
(55−11, 55, 361)-Net over F2 — Digital
Digital (44, 55, 361)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(255, 361, F2, 2, 11) (dual of [(361, 2), 667, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(255, 524, F2, 2, 11) (dual of [(524, 2), 993, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(255, 1048, F2, 11) (dual of [1048, 993, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(254, 1047, F2, 11) (dual of [1047, 993, 12]-code), using
- adding a parity check bit [i] based on linear OA(253, 1046, F2, 10) (dual of [1046, 993, 11]-code), using
- construction XX applied to C1 = C([1021,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(240, 1023, F2, 8) (dual of [1023, 983, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(251, 1023, F2, 11) (dual of [1023, 972, 12]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(230, 1023, F2, 6) (dual of [1023, 993, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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, 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 (see above)
- construction XX applied to C1 = C([1021,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- adding a parity check bit [i] based on linear OA(253, 1046, F2, 10) (dual of [1046, 993, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(254, 1047, F2, 11) (dual of [1047, 993, 12]-code), using
- OOA 2-folding [i] based on linear OA(255, 1048, F2, 11) (dual of [1048, 993, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(255, 524, F2, 2, 11) (dual of [(524, 2), 993, 12]-NRT-code), using
(55−11, 55, 4637)-Net in Base 2 — Upper bound on s
There is no (44, 55, 4638)-net in base 2, because
- 1 times m-reduction [i] would yield (44, 54, 4638)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 18019 603615 496489 > 254 [i]