Best Known (55, 81, s)-Nets in Base 5
(55, 81, 252)-Net over F5 — Constructive and digital
Digital (55, 81, 252)-net over F5, using
- 9 times m-reduction [i] based on digital (55, 90, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 45, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 45, 126)-net over F25, using
(55, 81, 508)-Net over F5 — Digital
Digital (55, 81, 508)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(581, 508, F5, 26) (dual of [508, 427, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 624, F5, 26) (dual of [624, 543, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(581, 624, F5, 26) (dual of [624, 543, 27]-code), using
(55, 81, 32087)-Net in Base 5 — Upper bound on s
There is no (55, 81, 32088)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 413 600564 304547 809346 710620 130956 021450 098838 994417 479265 > 581 [i]