Best Known (58, 58+26, s)-Nets in Base 5
(58, 58+26, 252)-Net over F5 — Constructive and digital
Digital (58, 84, 252)-net over F5, using
- 12 times m-reduction [i] based on digital (58, 96, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 48, 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, 48, 126)-net over F25, using
(58, 58+26, 625)-Net over F5 — Digital
Digital (58, 84, 625)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(584, 625, F5, 26) (dual of [625, 541, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 636, F5, 26) (dual of [636, 552, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(581, 625, F5, 26) (dual of [625, 544, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(573, 625, F5, 23) (dual of [625, 552, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(53, 11, F5, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(584, 636, F5, 26) (dual of [636, 552, 27]-code), using
(58, 58+26, 46524)-Net in Base 5 — Upper bound on s
There is no (58, 84, 46525)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 51706 463192 097798 545357 828068 553652 879663 879846 212786 363781 > 584 [i]