Best Known (51, 51+13, s)-Nets in Base 2
(51, 51+13, 174)-Net over F2 — Constructive and digital
Digital (51, 64, 174)-net over F2, using
- 21 times duplication [i] based on digital (50, 63, 174)-net over F2, using
- net defined by OOA [i] based on linear OOA(263, 174, F2, 13, 13) (dual of [(174, 13), 2199, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(263, 1045, F2, 13) (dual of [1045, 982, 14]-code), using
- construction XX applied to C1 = C([1021,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1021,10]) [i] based on
- 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(251, 1023, F2, 11) (dual of [1023, 972, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(261, 1023, F2, 13) (dual of [1023, 962, 14]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 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, using
- linear OA(21, 11, F2, 1) (dual of [11, 10, 2]-code) (see above)
- construction XX applied to C1 = C([1021,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1021,10]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(263, 1045, F2, 13) (dual of [1045, 982, 14]-code), using
- net defined by OOA [i] based on linear OOA(263, 174, F2, 13, 13) (dual of [(174, 13), 2199, 14]-NRT-code), using
(51, 51+13, 349)-Net over F2 — Digital
Digital (51, 64, 349)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(264, 349, F2, 3, 13) (dual of [(349, 3), 983, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(264, 1047, F2, 13) (dual of [1047, 983, 14]-code), using
- adding a parity check bit [i] based on linear OA(263, 1046, F2, 12) (dual of [1046, 983, 13]-code), using
- construction XX applied to C1 = C([1021,8]), C2 = C([1,10]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([1021,10]) [i] based on
- 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(250, 1023, F2, 10) (dual of [1023, 973, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(261, 1023, F2, 13) (dual of [1023, 962, 14]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [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(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([1021,8]), C2 = C([1,10]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([1021,10]) [i] based on
- adding a parity check bit [i] based on linear OA(263, 1046, F2, 12) (dual of [1046, 983, 13]-code), using
- OOA 3-folding [i] based on linear OA(264, 1047, F2, 13) (dual of [1047, 983, 14]-code), using
(51, 51+13, 4326)-Net in Base 2 — Upper bound on s
There is no (51, 64, 4327)-net in base 2, because
- 1 times m-reduction [i] would yield (51, 63, 4327)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 9 223553 892960 374101 > 263 [i]