Best Known (39−9, 39, s)-Nets in Base 2
(39−9, 39, 132)-Net over F2 — Constructive and digital
Digital (30, 39, 132)-net over F2, using
- net defined by OOA [i] based on linear OOA(239, 132, F2, 9, 9) (dual of [(132, 9), 1149, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(239, 132, F2, 8, 9) (dual of [(132, 8), 1017, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(239, 529, F2, 9) (dual of [529, 490, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(239, 531, F2, 9) (dual of [531, 492, 10]-code), using
- construction XX applied to C1 = C([509,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [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(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(219, 511, F2, 5) (dual of [511, 492, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([509,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(239, 531, F2, 9) (dual of [531, 492, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(239, 529, F2, 9) (dual of [529, 490, 10]-code), using
- appending kth column [i] based on linear OOA(239, 132, F2, 8, 9) (dual of [(132, 8), 1017, 10]-NRT-code), using
(39−9, 39, 207)-Net over F2 — Digital
Digital (30, 39, 207)-net over F2, using
- net defined by OOA [i] based on linear OOA(239, 207, F2, 9, 9) (dual of [(207, 9), 1824, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(239, 207, F2, 8, 9) (dual of [(207, 8), 1617, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(239, 207, F2, 2, 9) (dual of [(207, 2), 375, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(239, 265, F2, 2, 9) (dual of [(265, 2), 491, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(239, 530, F2, 9) (dual of [530, 491, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(239, 531, F2, 9) (dual of [531, 492, 10]-code), using
- construction XX applied to C1 = C([509,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [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(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(219, 511, F2, 5) (dual of [511, 492, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([509,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(239, 531, F2, 9) (dual of [531, 492, 10]-code), using
- OOA 2-folding [i] based on linear OA(239, 530, F2, 9) (dual of [530, 491, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(239, 265, F2, 2, 9) (dual of [(265, 2), 491, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(239, 207, F2, 2, 9) (dual of [(207, 2), 375, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(239, 207, F2, 8, 9) (dual of [(207, 8), 1617, 10]-NRT-code), using
(39−9, 39, 1597)-Net in Base 2 — Upper bound on s
There is no (30, 39, 1598)-net in base 2, because
- 1 times m-reduction [i] would yield (30, 38, 1598)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 275458 447599 > 238 [i]