Best Known (109−25, 109, s)-Nets in Base 5
(109−25, 109, 400)-Net over F5 — Constructive and digital
Digital (84, 109, 400)-net over F5, using
- 9 times m-reduction [i] based on digital (84, 118, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 59, 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, 59, 200)-net over F25, using
(109−25, 109, 3690)-Net over F5 — Digital
Digital (84, 109, 3690)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 3690, F5, 25) (dual of [3690, 3581, 26]-code), using
- 557 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 52 times 0, 1, 97 times 0, 1, 154 times 0, 1, 205 times 0) [i] based on linear OA(5100, 3124, F5, 25) (dual of [3124, 3024, 26]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using
- 557 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 52 times 0, 1, 97 times 0, 1, 154 times 0, 1, 205 times 0) [i] based on linear OA(5100, 3124, F5, 25) (dual of [3124, 3024, 26]-code), using
(109−25, 109, 2582438)-Net in Base 5 — Upper bound on s
There is no (84, 109, 2582439)-net in base 5, because
- 1 times m-reduction [i] would yield (84, 108, 2582439)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 3081 491308 104376 478045 401463 163136 116848 600308 656709 146529 873785 562030 181073 > 5108 [i]