Best Known (115−19, 115, s)-Nets in Base 5
(115−19, 115, 8687)-Net over F5 — Constructive and digital
Digital (96, 115, 8687)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (87, 106, 8681)-net over F5, using
- net defined by OOA [i] based on linear OOA(5106, 8681, F5, 19, 19) (dual of [(8681, 19), 164833, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5106, 78130, F5, 19) (dual of [78130, 78024, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5106, 78132, F5, 19) (dual of [78132, 78026, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5106, 78132, F5, 19) (dual of [78132, 78026, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5106, 78130, F5, 19) (dual of [78130, 78024, 20]-code), using
- net defined by OOA [i] based on linear OOA(5106, 8681, F5, 19, 19) (dual of [(8681, 19), 164833, 20]-NRT-code), using
- digital (0, 9, 6)-net over F5, using
(115−19, 115, 78169)-Net over F5 — Digital
Digital (96, 115, 78169)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5115, 78169, F5, 19) (dual of [78169, 78054, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
(115−19, 115, large)-Net in Base 5 — Upper bound on s
There is no (96, 115, large)-net in base 5, because
- 17 times m-reduction [i] would yield (96, 98, large)-net in base 5, but