Best Known (35−9, 35, s)-Nets in Base 2
(35−9, 35, 68)-Net over F2 — Constructive and digital
Digital (26, 35, 68)-net over F2, using
- net defined by OOA [i] based on linear OOA(235, 68, F2, 9, 9) (dual of [(68, 9), 577, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(235, 68, F2, 8, 9) (dual of [(68, 8), 509, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(217, 255, F2, 5) (dual of [255, 238, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- appending kth column [i] based on linear OOA(235, 68, F2, 8, 9) (dual of [(68, 8), 509, 10]-NRT-code), using
(35−9, 35, 128)-Net over F2 — Digital
Digital (26, 35, 128)-net over F2, using
- net defined by OOA [i] based on linear OOA(235, 128, F2, 9, 9) (dual of [(128, 9), 1117, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(235, 128, F2, 8, 9) (dual of [(128, 8), 989, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(235, 128, F2, 2, 9) (dual of [(128, 2), 221, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(235, 136, F2, 2, 9) (dual of [(136, 2), 237, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(235, 272, F2, 9) (dual of [272, 237, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(217, 255, F2, 5) (dual of [255, 238, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- OOA 2-folding [i] based on linear OA(235, 272, F2, 9) (dual of [272, 237, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(235, 136, F2, 2, 9) (dual of [(136, 2), 237, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(235, 128, F2, 2, 9) (dual of [(128, 2), 221, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(235, 128, F2, 8, 9) (dual of [(128, 8), 989, 10]-NRT-code), using
(35−9, 35, 795)-Net in Base 2 — Upper bound on s
There is no (26, 35, 796)-net in base 2, because
- 1 times m-reduction [i] would yield (26, 34, 796)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 17193 678606 > 234 [i]