Best Known (42, 42+10, s)-Nets in Base 2
(42, 42+10, 208)-Net over F2 — Constructive and digital
Digital (42, 52, 208)-net over F2, using
- net defined by OOA [i] based on linear OOA(252, 208, F2, 10, 10) (dual of [(208, 10), 2028, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(252, 1040, F2, 10) (dual of [1040, 988, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(252, 1044, F2, 10) (dual of [1044, 992, 11]-code), using
- 1 times truncation [i] based on linear OA(253, 1045, F2, 11) (dual of [1045, 992, 12]-code), using
- construction XX applied to C1 = C([1021,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [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(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(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [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([1021,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- 1 times truncation [i] based on linear OA(253, 1045, F2, 11) (dual of [1045, 992, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(252, 1044, F2, 10) (dual of [1044, 992, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(252, 1040, F2, 10) (dual of [1040, 988, 11]-code), using
(42, 42+10, 468)-Net over F2 — Digital
Digital (42, 52, 468)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(252, 468, F2, 2, 10) (dual of [(468, 2), 884, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(252, 522, F2, 2, 10) (dual of [(522, 2), 992, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(252, 1044, F2, 10) (dual of [1044, 992, 11]-code), using
- 1 times truncation [i] based on linear OA(253, 1045, F2, 11) (dual of [1045, 992, 12]-code), using
- construction XX applied to C1 = C([1021,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [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(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(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [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([1021,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- 1 times truncation [i] based on linear OA(253, 1045, F2, 11) (dual of [1045, 992, 12]-code), using
- OOA 2-folding [i] based on linear OA(252, 1044, F2, 10) (dual of [1044, 992, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(252, 522, F2, 2, 10) (dual of [(522, 2), 992, 11]-NRT-code), using
(42, 42+10, 3513)-Net in Base 2 — Upper bound on s
There is no (42, 52, 3514)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 4509 674578 807014 > 252 [i]