Best Known (51, 65, s)-Nets in Base 2
(51, 65, 132)-Net over F2 — Constructive and digital
Digital (51, 65, 132)-net over F2, using
- 21 times duplication [i] based on digital (50, 64, 132)-net over F2, using
- trace code for nets [i] based on digital (2, 16, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- trace code for nets [i] based on digital (2, 16, 33)-net over F16, using
(51, 65, 245)-Net over F2 — Digital
Digital (51, 65, 245)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(265, 245, F2, 2, 14) (dual of [(245, 2), 425, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(265, 265, F2, 2, 14) (dual of [(265, 2), 465, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(265, 530, F2, 14) (dual of [530, 465, 15]-code), using
- 1 times truncation [i] based on linear OA(266, 531, F2, 15) (dual of [531, 465, 16]-code), using
- construction XX applied to C1 = C([509,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([509,12]) [i] based on
- linear OA(255, 511, F2, 13) (dual of [511, 456, 14]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(255, 511, F2, 13) (dual of [511, 456, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(264, 511, F2, 15) (dual of [511, 447, 16]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(246, 511, F2, 11) (dual of [511, 465, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([509,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([509,12]) [i] based on
- 1 times truncation [i] based on linear OA(266, 531, F2, 15) (dual of [531, 465, 16]-code), using
- OOA 2-folding [i] based on linear OA(265, 530, F2, 14) (dual of [530, 465, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(265, 265, F2, 2, 14) (dual of [(265, 2), 465, 15]-NRT-code), using
(51, 65, 2099)-Net in Base 2 — Upper bound on s
There is no (51, 65, 2100)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 36 941682 642131 106196 > 265 [i]