Best Known (116−19, 116, s)-Nets in Base 5
(116−19, 116, 8691)-Net over F5 — Constructive and digital
Digital (97, 116, 8691)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- a shift-net [i]
- net from sequence [i] based on digital (1, 9)-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 (1, 10, 10)-net over F5, using
(116−19, 116, 78171)-Net over F5 — Digital
Digital (97, 116, 78171)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5116, 78171, F5, 19) (dual of [78171, 78055, 20]-code), using
- construction X with 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
- linear OA(5115, 78170, F5, 18) (dual of [78170, 78055, 19]-code), using Gilbert–Varšamov bound and bm = 5115 > Vbs−1(k−1) = 7 324985 381039 151986 424044 094291 832093 164735 069191 012855 217999 523671 794980 965157 [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(5115, 78169, F5, 19) (dual of [78169, 78054, 20]-code), using
- construction X with Varšamov bound [i] based on
(116−19, 116, large)-Net in Base 5 — Upper bound on s
There is no (97, 116, large)-net in base 5, because
- 17 times m-reduction [i] would yield (97, 99, large)-net in base 5, but