Best Known (109, 109+22, s)-Nets in Base 5
(109, 109+22, 7108)-Net over F5 — Constructive and digital
Digital (109, 131, 7108)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (98, 120, 7102)-net over F5, using
- net defined by OOA [i] based on linear OOA(5120, 7102, F5, 22, 22) (dual of [(7102, 22), 156124, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5120, 78122, F5, 22) (dual of [78122, 78002, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(5120, 78122, F5, 22) (dual of [78122, 78002, 23]-code), using
- net defined by OOA [i] based on linear OOA(5120, 7102, F5, 22, 22) (dual of [(7102, 22), 156124, 23]-NRT-code), using
- digital (0, 11, 6)-net over F5, using
(109, 109+22, 72522)-Net over F5 — Digital
Digital (109, 131, 72522)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5131, 72522, F5, 22) (dual of [72522, 72391, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5131, 78137, F5, 22) (dual of [78137, 78006, 23]-code), using
- (u, u+v)-construction [i] based on
- linear OA(511, 12, F5, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,5)), using
- dual of repetition code with length 12 [i]
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(511, 12, F5, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,5)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(5131, 78137, F5, 22) (dual of [78137, 78006, 23]-code), using
(109, 109+22, large)-Net in Base 5 — Upper bound on s
There is no (109, 131, large)-net in base 5, because
- 20 times m-reduction [i] would yield (109, 111, large)-net in base 5, but