Best Known (109−26, 109, s)-Nets in Base 5
(109−26, 109, 400)-Net over F5 — Constructive and digital
Digital (83, 109, 400)-net over F5, using
- 7 times m-reduction [i] based on digital (83, 116, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 58, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 58, 200)-net over F25, using
(109−26, 109, 3222)-Net over F5 — Digital
Digital (83, 109, 3222)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 3222, F5, 26) (dual of [3222, 3113, 27]-code), using
- 89 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 23 times 0, 1, 39 times 0) [i] based on 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]
- 89 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 23 times 0, 1, 39 times 0) [i] based on linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using
(109−26, 109, 1027867)-Net in Base 5 — Upper bound on s
There is no (83, 109, 1027868)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 15407 631101 887986 708525 540558 365468 703626 744768 182463 057865 084250 005342 343665 > 5109 [i]