Best Known (69, 91, s)-Nets in Base 5
(69, 91, 312)-Net over F5 — Constructive and digital
Digital (69, 91, 312)-net over F5, using
- 1 times m-reduction [i] based on digital (69, 92, 312)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (17, 28, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
- digital (41, 64, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 32, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- trace code for nets [i] based on digital (9, 32, 104)-net over F25, using
- digital (17, 28, 104)-net over F5, using
- (u, u+v)-construction [i] based on
(69, 91, 2888)-Net over F5 — Digital
Digital (69, 91, 2888)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(591, 2888, F5, 22) (dual of [2888, 2797, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(591, 3146, F5, 22) (dual of [3146, 3055, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(586, 3125, F5, 22) (dual of [3125, 3039, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(54, 20, F5, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(591, 3146, F5, 22) (dual of [3146, 3055, 23]-code), using
(69, 91, 743597)-Net in Base 5 — Upper bound on s
There is no (69, 91, 743598)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 4039 004332 895292 219553 883758 095052 485822 130150 479971 387219 323353 > 591 [i]