Best Known (70, 70+22, s)-Nets in Base 5
(70, 70+22, 356)-Net over F5 — Constructive and digital
Digital (70, 92, 356)-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 (42, 64, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 32, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 32, 126)-net over F25, using
- digital (17, 28, 104)-net over F5, using
(70, 70+22, 3131)-Net over F5 — Digital
Digital (70, 92, 3131)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(592, 3131, F5, 22) (dual of [3131, 3039, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(592, 3148, F5, 22) (dual of [3148, 3056, 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, 21, F5, 3) (dual of [21, 17, 4]-code or 21-cap in PG(3,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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 XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(592, 3148, F5, 22) (dual of [3148, 3056, 23]-code), using
(70, 70+22, 860758)-Net in Base 5 — Upper bound on s
There is no (70, 92, 860759)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 20195 029628 287669 084158 181177 978248 729726 126267 547180 095118 265525 > 592 [i]