Best Known (139, 139+32, s)-Nets in Base 4
(139, 139+32, 1126)-Net over F4 — Constructive and digital
Digital (139, 171, 1126)-net over F4, using
- 41 times duplication [i] based on digital (138, 170, 1126)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (26, 42, 98)-net over F4, using
- trace code for nets [i] based on digital (5, 21, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 21, 49)-net over F16, using
- digital (96, 128, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 32, 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, 32, 257)-net over F256, using
- digital (26, 42, 98)-net over F4, using
- (u, u+v)-construction [i] based on
(139, 139+32, 10336)-Net over F4 — Digital
Digital (139, 171, 10336)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4171, 10336, F4, 32) (dual of [10336, 10165, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 16401, F4, 32) (dual of [16401, 16230, 33]-code), using
- construction XX applied to Ce(32) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 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(32) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4171, 16401, F4, 32) (dual of [16401, 16230, 33]-code), using
(139, 139+32, 6164994)-Net in Base 4 — Upper bound on s
There is no (139, 171, 6164995)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 8 958980 797921 001977 337161 285390 270990 638154 504440 492841 297721 393855 457429 646736 112196 980073 586057 554912 > 4171 [i]