Best Known (51−16, 51, s)-Nets in Base 5
(51−16, 51, 208)-Net over F5 — Constructive and digital
Digital (35, 51, 208)-net over F5, using
- 1 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (9, 26, 104)-net over F25, using
(51−16, 51, 465)-Net over F5 — Digital
Digital (35, 51, 465)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(551, 465, F5, 16) (dual of [465, 414, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(551, 632, F5, 16) (dual of [632, 581, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(549, 625, F5, 16) (dual of [625, 576, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(545, 625, F5, 14) (dual of [625, 580, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(541, 625, F5, 13) (dual of [625, 584, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, 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(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(551, 632, F5, 16) (dual of [632, 581, 17]-code), using
(51−16, 51, 26882)-Net in Base 5 — Upper bound on s
There is no (35, 51, 26883)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 444111 434994 760773 634069 416431 964385 > 551 [i]