Best Known (31, 31+15, s)-Nets in Base 5
(31, 31+15, 142)-Net over F5 — Constructive and digital
Digital (31, 46, 142)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (23, 38, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 19, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 19, 66)-net over F25, using
- digital (1, 8, 10)-net over F5, using
(31, 31+15, 312)-Net over F5 — Digital
Digital (31, 46, 312)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(546, 312, F5, 15) (dual of [312, 266, 16]-code), using
- 1 times truncation [i] based on 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]
- 1 times truncation [i] based on linear OA(547, 313, F5, 16) (dual of [313, 266, 17]-code), using
(31, 31+15, 26312)-Net in Base 5 — Upper bound on s
There is no (31, 46, 26313)-net in base 5, because
- 1 times m-reduction [i] would yield (31, 45, 26313)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 28 426976 146991 797518 206627 907533 > 545 [i]