Best Known (28, 28+10, s)-Nets in Base 2
(28, 28+10, 58)-Net over F2 — Constructive and digital
Digital (28, 38, 58)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (7, 12, 36)-net over F2, using
- digital (16, 26, 29)-net over F2, using
- 3 times m-reduction [i] based on digital (16, 29, 29)-net over F2, using
(28, 28+10, 84)-Net over F2 — Digital
Digital (28, 38, 84)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(238, 84, F2, 10) (dual of [84, 46, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(238, 144, F2, 10) (dual of [144, 106, 11]-code), using
- construction XX applied to C1 = C({0,1,3,5,63}), C2 = C([1,7]), C3 = C1 + C2 = C([1,5]), and C∩ = C1 ∩ C2 = C({0,1,3,5,7,63}) [i] based on
- linear OA(229, 127, F2, 9) (dual of [127, 98, 10]-code), using the primitive cyclic code C(A) with length 127 = 27−1, defining set A = {0,1,3,5,63}, and minimum distance d ≥ |{−2,−1,…,6}|+1 = 10 (BCH-bound) [i]
- linear OA(228, 127, F2, 8) (dual of [127, 99, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(236, 127, F2, 11) (dual of [127, 91, 12]-code), using the primitive cyclic code C(A) with length 127 = 27−1, defining set A = {0,1,3,5,7,63}, and minimum distance d ≥ |{−2,−1,…,8}|+1 = 12 (BCH-bound) [i]
- linear OA(221, 127, F2, 6) (dual of [127, 106, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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, 8, F2, 1) (dual of [8, 7, 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({0,1,3,5,63}), C2 = C([1,7]), C3 = C1 + C2 = C([1,5]), and C∩ = C1 ∩ C2 = C({0,1,3,5,7,63}) [i] based on
- discarding factors / shortening the dual code based on linear OA(238, 144, F2, 10) (dual of [144, 106, 11]-code), using
(28, 28+10, 498)-Net in Base 2 — Upper bound on s
There is no (28, 38, 499)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 276308 478096 > 238 [i]