Best Known (110−21, 110, s)-Nets in Base 5
(110−21, 110, 1578)-Net over F5 — Constructive and digital
Digital (89, 110, 1578)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (76, 97, 1562)-net over F5, using
- net defined by OOA [i] based on linear OOA(597, 1562, F5, 21, 21) (dual of [(1562, 21), 32705, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(597, 15621, F5, 21) (dual of [15621, 15524, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(597, 15621, F5, 21) (dual of [15621, 15524, 22]-code), using
- net defined by OOA [i] based on linear OOA(597, 1562, F5, 21, 21) (dual of [(1562, 21), 32705, 22]-NRT-code), using
- digital (3, 13, 16)-net over F5, using
(110−21, 110, 15675)-Net over F5 — Digital
Digital (89, 110, 15675)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5110, 15675, F5, 21) (dual of [15675, 15565, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- linear OA(597, 15626, F5, 21) (dual of [15626, 15529, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(561, 15626, F5, 13) (dual of [15626, 15565, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(513, 49, F5, 7) (dual of [49, 36, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(513, 52, F5, 7) (dual of [52, 39, 8]-code), using
- a “LX†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(513, 52, F5, 7) (dual of [52, 39, 8]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
(110−21, 110, large)-Net in Base 5 — Upper bound on s
There is no (89, 110, large)-net in base 5, because
- 19 times m-reduction [i] would yield (89, 91, large)-net in base 5, but