Best Known (36, 55, s)-Nets in Base 4
(36, 55, 130)-Net over F4 — Constructive and digital
Digital (36, 55, 130)-net over F4, using
- 5 times m-reduction [i] based on digital (36, 60, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 30, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 30, 65)-net over F16, using
(36, 55, 185)-Net over F4 — Digital
Digital (36, 55, 185)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(455, 185, F4, 19) (dual of [185, 130, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(455, 255, F4, 19) (dual of [255, 200, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(455, 255, F4, 19) (dual of [255, 200, 20]-code), using
(36, 55, 5655)-Net in Base 4 — Upper bound on s
There is no (36, 55, 5656)-net in base 4, because
- 1 times m-reduction [i] would yield (36, 54, 5656)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 324 896444 908615 566273 442635 767962 > 454 [i]