Best Known (116−27, 116, s)-Nets in Base 5
(116−27, 116, 416)-Net over F5 — Constructive and digital
Digital (89, 116, 416)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (31, 44, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 22, 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, 22, 104)-net over F25, using
- digital (45, 72, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 36, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25 (see above)
- trace code for nets [i] based on digital (9, 36, 104)-net over F25, using
- digital (31, 44, 208)-net over F5, using
(116−27, 116, 3532)-Net over F5 — Digital
Digital (89, 116, 3532)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5116, 3532, F5, 27) (dual of [3532, 3416, 28]-code), using
- 392 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 37 times 0, 1, 66 times 0, 1, 104 times 0, 1, 147 times 0) [i] based on linear OA(5106, 3130, F5, 27) (dual of [3130, 3024, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(5106, 3125, F5, 27) (dual of [3125, 3019, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 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(26) ⊂ Ce(25) [i] based on
- 392 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 37 times 0, 1, 66 times 0, 1, 104 times 0, 1, 147 times 0) [i] based on linear OA(5106, 3130, F5, 27) (dual of [3130, 3024, 28]-code), using
(116−27, 116, 2160431)-Net in Base 5 — Upper bound on s
There is no (89, 116, 2160432)-net in base 5, because
- 1 times m-reduction [i] would yield (89, 115, 2160432)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 240 741832 164737 995694 846495 631136 013209 644816 169410 165199 410920 633596 327957 612225 > 5115 [i]