Best Known (100, 117, s)-Nets in Base 5
(100, 117, 48847)-Net over F5 — Constructive and digital
Digital (100, 117, 48847)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 18)-net over F5, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 4 and N(F) ≥ 18, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- digital (88, 105, 48829)-net over F5, using
- net defined by OOA [i] based on linear OOA(5105, 48829, F5, 17, 17) (dual of [(48829, 17), 829988, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5105, 390633, F5, 17) (dual of [390633, 390528, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(5105, 390633, F5, 17) (dual of [390633, 390528, 18]-code), using
- net defined by OOA [i] based on linear OOA(5105, 48829, F5, 17, 17) (dual of [(48829, 17), 829988, 18]-NRT-code), using
- digital (4, 12, 18)-net over F5, using
(100, 117, 390679)-Net over F5 — Digital
Digital (100, 117, 390679)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5117, 390679, F5, 17) (dual of [390679, 390562, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5114, 390674, F5, 17) (dual of [390674, 390560, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(16) ⊂ Ce(10) [i] based on
- linear OA(5114, 390676, F5, 15) (dual of [390676, 390562, 16]-code), using Gilbert–Varšamov bound and bm = 5114 > Vbs−1(k−1) = 5939 541861 136056 452280 740294 586875 940195 762091 848138 921475 198409 234928 441661 [i]
- linear OA(51, 3, F5, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(5114, 390674, F5, 17) (dual of [390674, 390560, 18]-code), using
- construction X with Varšamov bound [i] based on
(100, 117, large)-Net in Base 5 — Upper bound on s
There is no (100, 117, large)-net in base 5, because
- 15 times m-reduction [i] would yield (100, 102, large)-net in base 5, but