Best Known (32, 32+15, s)-Nets in Base 5
(32, 32+15, 144)-Net over F5 — Constructive and digital
Digital (32, 47, 144)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-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 (2, 9, 12)-net over F5, using
(32, 32+15, 344)-Net over F5 — Digital
Digital (32, 47, 344)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(547, 344, F5, 15) (dual of [344, 297, 16]-code), using
- 31 step Varšamov–Edel lengthening with (ri) = (1, 30 times 0) [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 step Varšamov–Edel lengthening with (ri) = (1, 30 times 0) [i] based on linear OA(546, 312, F5, 15) (dual of [312, 266, 16]-code), using
(32, 32+15, 33115)-Net in Base 5 — Upper bound on s
There is no (32, 47, 33116)-net in base 5, because
- 1 times m-reduction [i] would yield (32, 46, 33116)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 142 131558 145424 287414 696392 449265 > 546 [i]