Best Known (75−16, 75, s)-Nets in Base 4
(75−16, 75, 1042)-Net over F4 — Constructive and digital
Digital (59, 75, 1042)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (48, 64, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
- digital (3, 11, 14)-net over F4, using
(75−16, 75, 3056)-Net over F4 — Digital
Digital (59, 75, 3056)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(475, 3056, F4, 16) (dual of [3056, 2981, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(475, 4111, F4, 16) (dual of [4111, 4036, 17]-code), using
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(473, 4096, F4, 17) (dual of [4096, 4023, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(461, 4096, F4, 14) (dual of [4096, 4035, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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, 14, F4, 1) (dual of [14, 13, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(475, 4111, F4, 16) (dual of [4111, 4036, 17]-code), using
(75−16, 75, 553192)-Net in Base 4 — Upper bound on s
There is no (59, 75, 553193)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1427 255258 980069 184059 255410 270867 714059 168725 > 475 [i]