Best Known (31−8, 31, s)-Nets in Base 2
(31−8, 31, 71)-Net over F2 — Constructive and digital
Digital (23, 31, 71)-net over F2, using
- net defined by OOA [i] based on linear OOA(231, 71, F2, 8, 8) (dual of [(71, 8), 537, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(231, 71, F2, 7, 8) (dual of [(71, 7), 466, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 7, 4) (dual of [(32, 7), 214, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- linear OOA(221, 39, F2, 7, 8) (dual of [(39, 7), 252, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 21, 39)-net over F2, using
- linear OOA(210, 32, F2, 7, 4) (dual of [(32, 7), 214, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(231, 71, F2, 7, 8) (dual of [(71, 7), 466, 9]-NRT-code), using
(31−8, 31, 89)-Net over F2 — Digital
Digital (23, 31, 89)-net over F2, using
- net defined by OOA [i] based on linear OOA(231, 89, F2, 8, 8) (dual of [(89, 8), 681, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(231, 89, F2, 7, 8) (dual of [(89, 7), 592, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(231, 89, F2, 8) (dual of [89, 58, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(231, 144, F2, 8) (dual of [144, 113, 9]-code), using
- construction XX applied to C1 = C({0,1,3,63}), C2 = C([1,5]), C3 = C1 + C2 = C([1,3]), and C∩ = C1 ∩ C2 = C({0,1,3,5,63}) [i] based on
- linear OA(222, 127, F2, 7) (dual of [127, 105, 8]-code), using the primitive cyclic code C(A) with length 127 = 27−1, defining set A = {0,1,3,63}, and minimum distance d ≥ |{−2,−1,…,4}|+1 = 8 (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(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(214, 127, F2, 4) (dual of [127, 113, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 5 [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({0,1,3,63}), C2 = C([1,5]), C3 = C1 + C2 = C([1,3]), and C∩ = C1 ∩ C2 = C({0,1,3,5,63}) [i] based on
- discarding factors / shortening the dual code based on linear OA(231, 144, F2, 8) (dual of [144, 113, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(231, 89, F2, 8) (dual of [89, 58, 9]-code), using
- appending kth column [i] based on linear OOA(231, 89, F2, 7, 8) (dual of [(89, 7), 592, 9]-NRT-code), using
(31−8, 31, 470)-Net in Base 2 — Upper bound on s
There is no (23, 31, 471)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2147 554331 > 231 [i]