Best Known (88, 88+21, s)-Nets in Base 5
(88, 88+21, 1574)-Net over F5 — Constructive and digital
Digital (88, 109, 1574)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 12, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-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 (2, 12, 12)-net over F5, using
(88, 88+21, 15668)-Net over F5 — Digital
Digital (88, 109, 15668)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 15668, F5, 21) (dual of [15668, 15559, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5108, 15666, F5, 21) (dual of [15666, 15558, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] 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]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(511, 41, F5, 6) (dual of [41, 30, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(511, 45, F5, 6) (dual of [45, 34, 7]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(5108, 15667, F5, 20) (dual of [15667, 15559, 21]-code), using Gilbert–Varšamov bound and bm = 5108 > Vbs−1(k−1) = 113 145956 560424 515557 002505 605868 992358 850643 891009 891679 054137 181742 766425 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(5108, 15666, F5, 21) (dual of [15666, 15558, 22]-code), using
- construction X with Varšamov bound [i] based on
(88, 88+21, large)-Net in Base 5 — Upper bound on s
There is no (88, 109, large)-net in base 5, because
- 19 times m-reduction [i] would yield (88, 90, large)-net in base 5, but