Best Known (52−17, 52, s)-Nets in Base 5
(52−17, 52, 208)-Net over F5 — Constructive and digital
Digital (35, 52, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 26, 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
(52−17, 52, 333)-Net over F5 — Digital
Digital (35, 52, 333)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(552, 333, F5, 17) (dual of [333, 281, 18]-code), using
- 15 step Varšamov–Edel lengthening with (ri) = (1, 14 times 0) [i] based on linear OA(551, 317, F5, 17) (dual of [317, 266, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(551, 313, F5, 17) (dual of [313, 262, 18]-code), using an extension Ce(16) of the narrow-sense BCH-code C(I) with length 312 | 54−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(547, 313, F5, 16) (dual of [313, 266, 17]-code), using an extension Ce(15) of the narrow-sense BCH-code C(I) with length 312 | 54−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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
- 15 step Varšamov–Edel lengthening with (ri) = (1, 14 times 0) [i] based on linear OA(551, 317, F5, 17) (dual of [317, 266, 18]-code), using
(52−17, 52, 26882)-Net in Base 5 — Upper bound on s
There is no (35, 52, 26883)-net in base 5, because
- 1 times m-reduction [i] would yield (35, 51, 26883)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 444111 434994 760773 634069 416431 964385 > 551 [i]