Best Known (131−19, 131, s)-Nets in Base 5
(131−19, 131, 43413)-Net over F5 — Constructive and digital
Digital (112, 131, 43413)-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 (102, 121, 43403)-net over F5, using
- net defined by OOA [i] based on linear OOA(5121, 43403, F5, 19, 19) (dual of [(43403, 19), 824536, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5121, 390628, F5, 19) (dual of [390628, 390507, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5121, 390633, F5, 19) (dual of [390633, 390512, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 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(5121, 390633, F5, 19) (dual of [390633, 390512, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5121, 390628, F5, 19) (dual of [390628, 390507, 20]-code), using
- net defined by OOA [i] based on linear OOA(5121, 43403, F5, 19, 19) (dual of [(43403, 19), 824536, 20]-NRT-code), using
- digital (1, 10, 10)-net over F5, using
(131−19, 131, 390676)-Net over F5 — Digital
Digital (112, 131, 390676)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5131, 390676, F5, 19) (dual of [390676, 390545, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5130, 390674, F5, 19) (dual of [390674, 390544, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(59, 49, F5, 5) (dual of [49, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(5130, 390675, F5, 18) (dual of [390675, 390545, 19]-code), using Gilbert–Varšamov bound and bm = 5130 > Vbs−1(k−1) = 5 554583 899118 143196 584302 708100 352978 795203 033577 510806 938574 584992 774560 037859 667709 645273 [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(5130, 390674, F5, 19) (dual of [390674, 390544, 20]-code), using
- construction X with Varšamov bound [i] based on
(131−19, 131, large)-Net in Base 5 — Upper bound on s
There is no (112, 131, large)-net in base 5, because
- 17 times m-reduction [i] would yield (112, 114, large)-net in base 5, but