Best Known (139, 139+36, s)-Nets in Base 4
(139, 139+36, 1058)-Net over F4 — Constructive and digital
Digital (139, 175, 1058)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 31, 30)-net over F4, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- F4 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- digital (108, 144, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 36, 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, 36, 257)-net over F256, using
- digital (13, 31, 30)-net over F4, using
(139, 139+36, 4776)-Net over F4 — Digital
Digital (139, 175, 4776)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4175, 4776, F4, 36) (dual of [4776, 4601, 37]-code), using
- 668 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 38 times 0, 1, 60 times 0, 1, 88 times 0, 1, 118 times 0, 1, 145 times 0, 1, 168 times 0) [i] based on linear OA(4162, 4095, F4, 36) (dual of [4095, 3933, 37]-code), using
- 1 times truncation [i] based on linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 1 times truncation [i] based on linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using
- 668 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 38 times 0, 1, 60 times 0, 1, 88 times 0, 1, 118 times 0, 1, 145 times 0, 1, 168 times 0) [i] based on linear OA(4162, 4095, F4, 36) (dual of [4095, 3933, 37]-code), using
(139, 139+36, 1796248)-Net in Base 4 — Upper bound on s
There is no (139, 175, 1796249)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2293 519183 359380 781890 616367 864516 998065 335819 097928 423245 996225 930638 148133 932258 271352 390072 804202 219572 > 4175 [i]