Best Known (37, 37+16, s)-Nets in Base 4
(37, 37+16, 240)-Net over F4 — Constructive and digital
Digital (37, 53, 240)-net over F4, using
- 1 times m-reduction [i] based on digital (37, 54, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 18, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 18, 80)-net over F64, using
(37, 37+16, 300)-Net over F4 — Digital
Digital (37, 53, 300)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 300, F4, 16) (dual of [300, 247, 17]-code), using
- 39 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 17 times 0) [i] based on linear OA(448, 256, F4, 16) (dual of [256, 208, 17]-code), using
- 1 times truncation [i] based on linear OA(449, 257, F4, 17) (dual of [257, 208, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 257 | 48−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(449, 257, F4, 17) (dual of [257, 208, 18]-code), using
- 39 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 17 times 0) [i] based on linear OA(448, 256, F4, 16) (dual of [256, 208, 17]-code), using
(37, 37+16, 12217)-Net in Base 4 — Upper bound on s
There is no (37, 53, 12218)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 81 134070 340263 332204 451599 903875 > 453 [i]