Best Known (32, 32+10, s)-Nets in Base 2
(32, 32+10, 77)-Net over F2 — Constructive and digital
Digital (32, 42, 77)-net over F2, using
- (u, u+v)-construction [i] based on
(32, 32+10, 136)-Net over F2 — Digital
Digital (32, 42, 136)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(242, 136, F2, 2, 10) (dual of [(136, 2), 230, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(242, 272, F2, 10) (dual of [272, 230, 11]-code), using
- 1 times truncation [i] based on linear OA(243, 273, F2, 11) (dual of [273, 230, 12]-code), using
- construction XX applied to C1 = C([253,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- 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(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(241, 255, F2, 11) (dual of [255, 214, 12]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [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(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([253,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- 1 times truncation [i] based on linear OA(243, 273, F2, 11) (dual of [273, 230, 12]-code), using
- OOA 2-folding [i] based on linear OA(242, 272, F2, 10) (dual of [272, 230, 11]-code), using
(32, 32+10, 873)-Net in Base 2 — Upper bound on s
There is no (32, 42, 874)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 4 422213 758546 > 242 [i]