Best Known (74−15, 74, s)-Nets in Base 4
(74−15, 74, 1062)-Net over F4 — Constructive and digital
Digital (59, 74, 1062)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 34)-net over F4, using
- trace code for nets [i] based on digital (0, 7, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- trace code for nets [i] based on digital (0, 7, 17)-net over F16, using
- digital (45, 60, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
- digital (7, 14, 34)-net over F4, using
(74−15, 74, 4164)-Net over F4 — Digital
Digital (59, 74, 4164)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(474, 4164, F4, 15) (dual of [4164, 4090, 16]-code), using
- 49 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0) [i] based on linear OA(468, 4109, F4, 15) (dual of [4109, 4041, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(467, 4096, F4, 15) (dual of [4096, 4029, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(455, 4096, F4, 13) (dual of [4096, 4041, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- 49 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0) [i] based on linear OA(468, 4109, F4, 15) (dual of [4109, 4041, 16]-code), using
(74−15, 74, 2140040)-Net in Base 4 — Upper bound on s
There is no (59, 74, 2140041)-net in base 4, because
- 1 times m-reduction [i] would yield (59, 73, 2140041)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 89 203209 405597 240380 611233 142617 634838 065768 > 473 [i]